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UNIVERSITY  (JF  TRANSYLVANIA 


THE  NOTION  OF  NUMBER  AND 
THE  NOTION  OF  GLASS 


BY 


RICHARD  A.  ARMS 


A  THESIS 

PRESENTED   TO   THE  FACULTY   OF  THE   GRADUATE  SCHOOL  IN 

PARTIAL    FULFILLMENT    OF    TFE    REQUIREMENTS    FOR 

THE    DEGREE    OF    DOCTOR    OF    PHILOSOPHY 


PHILADELPHIA 
1917 


1 


UNIVERSITY  OF  PENNSYLVANIA 


THE  NOTION  OF  NUMBER  AND 
THE  NOTION  OF  CLASS 


BY 

RICHARD  A.  ARMS 


A  THESIS 

PRESENTED   TO   THE   FACULTY   OF  THE   GRADUATE   SCHOOL  IN 

PARTIAL    FULFILLMENT    OF    THE    REQUIREMENTS    FOR 

THE    DEGREE    OF    DOCTOR    OF    PHILOSOPHY 


PHILADELPHIA 
1917 


Press  of 

Steinman  &  Foltz. 

Lancaster.  Pa. 


CONTENTS 

Page 

A.  The  Notion  of  Number 5 

B.  The  Psychological  Aspect 19 

C.  The  Notion  of  Class 39 

D.  Mathematical  Usage 55 


The  Notion  of  Number  and  the  Notion 

of  Class 

A.    The  Notion  of  Number 

§i.  Introductory 

Discussions  about  the  nature  of  number  may  be  readily 
divided  into  various  types.  It  is  one  of  the  striking  character- 
istics of  the  true  mathematician  that  anything  in  the  way  of  a 
vague  statement  causes  him  acute  discomfort.  He  desires  a 
change  from  the  suggestive  impressionistic  expression  of  the 
subjectively  self-evident  to  an  objective  explicit  derivation. 
A  striking  example  of  this  is  Frege's  rebellion  against  psycholog- 
ical theories  of  number.  "States  of  mind  have  no  place  in 
mathematics,"  says  the  mathematician,  and,  so  far  from  differing 
with  him,  we  may  well  inquire  whether,  as  such,  they  have  any 
place  in  philosophy. 

To  the  post-Kantian  metaphysician  no  such  ideal  ever  pre- 
sented itself.  For  him  the  boasted  Copernican  revolution  had 
indeed  taken  place,  and  like  our  pragmatists,  he  could  think 
of  no  problem  except  in  connection  with  some  mind.  Number 
depends  upon  the  way  we  think  of  it  and  the  logically  simple 
is  what  we  can  think  of  most  readily.  The  method  used  is 
introspection.  Opposed  to  these  is  the  empiricist  who  demands 
that  all  be  reduced  to  physical  terms.  Crudities  such  as  this 
represent  J.  S.  Mill's  rash  intrusion  into  the  philosophy  of  arith- 
metic. 

Corresponding  to  these  three  classes  of  thinkers  we  find  widely 
different  conceptions  of  number  and  three  variants  of  Aris- 
totelian logic.  The  empiricist  defines  number  as  a  physical 
property  of  external  objects  and  emphasizes  induction;  the 
Kantian  talks  about  thought-content,  introducing  all  manner 
of  the  psychology  of  reasoning  into  his  so-called  logic.  It 
remains  for  the  formalistic  mathematician  to  regard  number 
as  definable  in  terms  of  class  and  that  bewildering  maze  of 
svmbolic  ramifications  which  has  gained  the  title  of  logistic 
as  the  pinnacle  of  mathematical  method. 

5 


6  The  Notion  of  Number  and  the  Notion  of  Class 

The  present  discussion  will  not  be  confined  to  a  purely  mathe- 
matical treatment  of  the  notion  of  number.  The  objects  of 
arithmetic  are  not  entities  met  with  only  on  the  pages  of  text- 
books; number  is  something  which  we  use  in  every  variety  of 
present  day  experience,  and  which  even  primitive  man  was 
forced  to  recognize.  Some  introduction  of  mental  states  seems 
unavoidable.  However,  we  may  demand  that  this  introduction 
be  made  in  terms  of  actual  or  virtual  behavior  so  that  any  descrip- 
tion of  man's  relation  to  number  can  be  verified.  The  result 
must  be  fixed  and  accessible. 

We  shall  first  be  concerned  with  the  notion  of  number  as 
it  appeared  to  Mill  and  the  Kantians  and  shall  attempt  to  show 
that  the  various  historical  positions  are  liable  to  one  or  both 
of  the  following  objections:  (a)  subjective  mental  states  have 
been  introduced,  (b)  violence  has  been  done  to  the  facts.  Frege's 
distinction  here  between  a  posteriori  and  a  priori  methods  is 
important.  He  holds  that  a  priori  is  deductive,  a  posteriori, 
inductive;  synthetic  means  that  which  uses  conceptions  belong- 
ing to  a  special  science  while  an  analytic  proposition  is  derived 
from  pure  logic  alone.  Frege  held  that  the  concepts  and  laws 
of  arithmetic  are  analytic  a  priori,  that  they  are  derived  from 
the  fundamental  definitions  and  axioms  of  logic.  We  shall 
find  Bertrand  Russell  espousing  the  same  cause  although  inde- 
pendently and  from  slightly  different  motives. 

§2.  Empirical  and  Kantian  Views 

John  Stuart  Mill  is  the  classic  exponent  of  the  inductive 
theory  of  number  and  his  conception  of  the  basic  truths  of 
arithmetic  is  narrow  in  the  extreme.  We  find  him  contending 
that  number  is  the  expression  of  an  act,  a  physical  property 
of  physical  objects  on  the  same  plane  as  weight,  color,  and 
extension;  that  the  laws  of  arithmetic  are  merely  inductions 
of  a  very  high  order.  Addition  is  physical  combination  of 
objects,  and  calculation  in  general  does  not  depend  upon  rule 
and  definition  but  upon  observation  of  actual  objects. 

Leibnitz,  Locke,  and  commonsense  agree,  however,  that  all 
entities  whatever  can  be  numbered.  To  render  his  doctrine 
consistent  the  empiricist  must  hold  that  all  existents  are  physical 
beings.  Here  we  may  reply  that  if  this  is  the  case,  we  are 
confined   to  the  mechanical  world  order  and   that  where  the 


The  Notion  of  Number  7 

empiricist  desired  to  put  number  in  a  concrete  world,  rich  with 
individuality,  he  either  contradicted  himself,  or  placed  number 
in  the  most  abstract  of  all  worlds, — a  world  from  which  life, 
mind,  and  purpose  had  been  perforce  abstracted. 

Kant's  theory  of  the  "Schema"  of  Quantity  is  noteworthy 
because  of  the  manifest  belief  that  number  is  the  science  of 
time,  while  geometry  is  the  science  of  space  and  the  emphasis 
on  succession,  which  is  due  to  an  introspective  analysis  of  count- 
ing. It  must  always  remain  a  matter  of  doubt  whether  Kant 
believed  that  arithmetic  was  the  science  of  time  from  the  merits 
of  the  case  or  because  of  the  symmetry  of  his  system.  We  may 
not  without  grounds  suspect  him  of  falling  victim  to  the  archi- 
tectonic passion. 

Among  the  closer  followers  of  Kant,  in  their  emphasis  on  the 
intimate  connection  of  number  and  time  are  Bain,  Hamilton, 
and  Helmholtz.  The  most  interesting  of  these  theories  is  found 
in  Helmholtz'  "Zahlen  und  Messen."  It  offers  the  paradox 
of  combining  in  its  attitude  the  extremes  of  both  the  objective 
and  subjective  standpoints  and  is  a  fine  example  of  nominalism, 
regarding  numbers  as  mere  words  or  symbols  which  have  no 
significance  except  when  used.  Berkeley  was  among  the  first 
to  regard  number  as  a  mere  tag.  This  standpoint  must  be 
sharply  differentiated  from  that  of  the  formalists  in  mathe- 
matics. The  mathematician  at  the  close  of  his  manipulations 
is  concerned  with  the  various  interpretations  of  his  symbols; 
the  nominalist  says  in  effect  that  there  is  no  interpretation, 
that  mathematics  deals  with  words  alone.  This  sweeping 
conclusion  cannot  be  accepted  without  more  reason  than  has 
yet  appeared. 

Lange,  Baumann,  and  Brix  cling  to  the  Kantian  formula  of 
a  "Schematismus"  but  are  inclined  to  hold  that  number  is  more 
closely  connected  with  the  intuitions  of  space  than  those  of 
time.  Lange  in  his  "Logische  Studien"  blends  in  a  curious 
way  the  views  of  Kant  and  Mill.  He  agrees  with  Mill  in  deriv- 
ing number  from  the  contemplation  of  spatial  objects  and  follows 
Kant  in  his  reiteration  of  the  magic  word  "Synthesis."  He  even 
goes  so  far  as  to  say  that  all  logical  and  mathematical  thinking 
is  in  terms  of  spatial  syntheses  which  are  thus  not  only  the 
basis  of  the  number  concept,  but  also  of  everything  else.  How- 
ever, such  terms  as  "Schema"  and  "Synthesis"  tend  to  obscure 


8  The  Notion  of  Number  and  the  Notion  of  Class 

more  than  they  explain  and  are  more  puzzling  than  the  notion 
of  number  itself. 

Let  us  pause  for  a  moment  to  estimate  the  value  of  the  intro- 
spective method.  The  theories  above  cited  have  been  gained 
by  its  use.  As  a  result  we  see  one  party  holding  that  number 
depends  upon  time  and  in  no  way  involves  space;  another  school, 
whose  members  happen  to  be  of  the  visual  type,  contending 
for  the  opposite  theory.  Such  a  divergence  casts  no  doubt 
upon  the  validity  of  the  theorems  of  arithmetic.  It  is  rather 
upon  the  method  that  these  inconsistent  conclusions  reflect. 
Is  it  not  by  this  time  clear  that  whatever  an  investigator  dis- 
covers by  delving  within  the  recesses  of  his  "inner  experience" 
cannot  be  safely  offered  as  a  truth  unless  it  is  solidly  verified  by 
observable  fact?  Otherwise  we  have  the  mad  scramble  of  each 
man  to  his  taste,  and  as  many  philosophies  of  number  as  there 
are  philosophers. 

§3.     Number  as  Derived  from  Identity  and  Difference 

Jevons  preferred  to  assume  a  higher  point  of  view.  He  con- 
tends that  every  means  of  differentiation  can  be  a  source  of 
plurality,  that  number  is  only  another  name  for  difference. 
However,  it  is  clear  that  a  certain  likeness  as  well  as  difference 
must  be  possessed  by  the  counted  units.  Now  if  the  units  of 
arithmetic  are  radically  different,  the  fundamental  laws  of  the 
subject  must  be  stated  in  terms  of  any  unit  whatever,  or  we  will 
be  quite  unable  to  calculate.  However,  it  is  manifest  that  when 
we  say  "any  unit"  it  is  the  quality  of  being  a  unit  which  is  essen- 
tial and  their  likeness  is  more  important  than  their  difference. 

Hegel's  dialectic  treatment  of  number  in  the  "  Logik"  leaves 
much  to  be  desired  from  a  standpoint  of  clarity  but  is  extremely 
suggestive.  He  shows  that  the  contradictions  in  the  nature 
of  One  make  One  and  not-One  qualitatively  identical  and  thus 
engender  a  Plurality.  The  synthesis  of  unity  and  plurality 
is  number.  This  synthesis  denotes  the  collective  unity  of  a 
class  whose  reality  lies  in  its  parts.  The  units  composing  the 
number,  although  distinct,  are  qualitatively  identical.  Thus 
from  the  discontinuity  or  isolation  of  the  units  we  pass  to  their 
continuity  or  connection.  In  realizing  this  fact,  Hegel  has 
advanced  beyond  Jevons. 

Schuppe  in  his  "Grundriss  der  Erkenntnis  theorie  und  Logik" 
sets  forth  the  theory  that  plurality  is  already  presupposed  in 


The  Notion  of  Number  9 

the  principle  of  identity,  for  the  concepts  of  identity  and  differ- 
ence presuppose  at  least  two  distinct  impressions.  For  number 
the  principal  matter  is  the  comprehension  ("Zusammenfassung") 
of  the  units,  by  which  a  pre-existing  difference  is  made  explicit. 
Number  also  asserts  likeness  and  needs  another  concept  under 
which  the  differentiated  objects  are  to  be  put,  for  only  the 
similar  can  be  counted.  So  far,  Schuppe's  analysis  is  accurate 
and  subtle.  However,  he  goes  on  to  say  that  difference  in 
spatial  position  is  essential  to  number  and  that  when  we  seem 
to  count  non-spatial  objects  we  really  localize  them  in  imaginary 
space, — being  apparently  convinced  that  because  he,  Schuppe, 
happens  to  visualize  counted  objects,  every  one  else  does  so. 

Bergson's  views,  as  elaborated  in  the  second  chapter  of  "Time 
and  Free  Will,"  are  not  unlike  those  of  Schuppe.  He  insists 
that  psychic  states  are  continuous  and  non-spatial,  whereas 
number  is  discrete  and  spatial.  It  is  not  enough  to  say  that 
number  deals  with  units.  We  also  require  that  these  units  be 
identical  with  one  another,  so  far  as  the  counting  is  concerned. 
Nevertheless,  the  units  must  be  distinct.  We  conclude,  there- 
fore, that  they  are  differentiated  by  space. 

Everything  is  not  counted  in  the  same  way,  for  there  are 
two  different  kinds  of  multiplicity.  Material  objects  are  given 
as  separated  by  space.  Psychic  states,  however,  cannot  be 
retained  in  time;  they  must  be  differentiated  with  respect  to 
some  homogeneous  medium.  Their  continuity  is  not  numerical 
but  a  succession  of  qualitative  changes  which  melt  into  one 
another.  Between  the  discrete  and  the  continuous  there  is  a 
wide  gulf. 

A  correspondingly  sharp  line  of  demarcation  between  number 
and  the  continuum  is  found  in  Sigward's  "Logik."  Number, 
to  him,  is  only  a  development  of  the  notions  of  Identity  and 
Difference.  Everything  we  take  to  be  one  is  cut  out  of  the 
continuum  by  a  definite  limited  act  of  perception,  whether  that 
continuum  be  time  or  space.  A  number  is  not  a  bare  plurality 
but  a  plurality  definitely  comprehended  and  bounded,  and  its 
possibility  lies  in  the  fact  that  we  are  conscious  of  our  progress 
from  one  unit  to  another,  making  use  of  formal  activities  alone. 
Number  is  a  free  creation  of  our  self-conscious  thought  and  is 
independent  of  what  is  given  by  sense. 

Now  the  introduction  of  any  desired  group  of  fractions  and 
irrationals  between  two  consecutive  integers  seems  to  reduce 


io  The  Notion  of  Number  and  the  Notion  of  Class 

number,  originally  discrete,  to  a  continuum.  Notwithstanding 
the  superficial  plausibility  of  this  conclusion,  argues  Sigwart, 
number  can  never  be  anything  but  discrete;  since,  however  far 
we  push  our  interpolations,  we  can  never  succeed  in  reaching 
more  than  a  finite  number  of  intervening  members.  The  con- 
tinuous process  which  intuition  gives  us  in  space  and  time  can 
never  be  expressed  in  the  forms  of  number. 

Here  is  evidently  a  point  of  view  -closely  akin  to  that  of  Berg- 
son.  Number  and  the  continuum  are  placed  as  far  asunder 
as  the  poles  and  "never  the  twain  shall  meet."  Sigwart's  spirit 
in  this  discussion  is  not  far  from  that  of  Bergson,  when  the 
latter  accuses  physics  of  transforming  continuous  movement 
into  mathematical  smoke.  The  continuum  is  an  insurmountable 
obstacle  and  barrier  to  logical  progress.  But  was  it  not  Hegel 
himself  who  said  "Knowledge  of  a  limit  is  knowledge  beyond"? 

Bosanquet  is  rather  a  failure  as  a  philosopher  of  the  continuum. 
He  makes  continuity  an  invariable  concomitant  of  classification 
and  accordingly  must  hold  that  a  dozen  eggs  form  just  as  con- 
tinuous a  group  as  the  points  of  a  line.  The  problem  as  it  arose 
is  not  attacked.  As  a  matter  of  fact  the  only  consistent  answer 
Bosanquet  could  have  given  is  one  like  Sigwart's,  for  their 
theory  of  counting  is  similar  in  all  essentials. 

The  results  of  these  two  self-styled  logicians  are  important 
for  two  reasons.  They  continually  emphasize  the  abstract 
nature  of  numbering  and  make  it  subordinate  to  the  logical 
process  of  classification;  and  they  are  aware  that  order  is  a  funda- 
mental notion.  While  their  methods  are  those  of  subjective 
psychology,  these  conclusions  are  not  necessarily  so.  They  lead 
to  the  notion  that  number  is  derived  from  formal  logic,  a  position 
most  prominently  represented  by  Frege  and  Russell. 

§4.     Number  as  Derived  from  Formal  Logic 

To  gain  his  -definition  of  number  Frege  introduces  a  new 
and  widely  applicable  logical  method.  Just  as  we  say  that 
lines  parallel  to  a  given  line  have  the  same  direction,  so  we  lay 
it  down  that  concepts  having  the  same  number  are  those  which 
are  equivalent  ("gleichzahlig")  to  a  given  concept  A  and  the 
particular  number  is  defined  as  the  "Umfang"  of  the  concept 
"equivalent  to  A."  The  relation  of  equivalence  is  that  of  one- 
one  correspondence,  and  Frege  gives  the  following  definition  of 


The  Notion  of  Number  n 

it  in  purely  logical  terms:  Two  concepts  F  and  G  bear  to  each 
other  the  relation  of  one- one  correspondence  when  there  is  a 
relation  R  such  that  if  a,e  belong  to  F  and  b,d  to  G,   then, 
(i)  IfdRa  and  d  R  e  then  a  and  e  are  identical. 
(2)  If  d  R  a  and  b  R  a  then  d  and  b  are  identical. 
The  word  "one"  is  dexterously  avoided. 

The  number  Zero  offers  some  peculiar  difficulties.  It  is 
defined  as  the  number  which  pertains  to  the  concept  "non- 
identical  with  itself."  The  question  naturally  arises, — how  can 
two  null-concepts  be  placed  in  a  one-one  relation  if  there  is 
nothing  there  to  correspond?  Frege's  device  is  the  clever  but 
unconvincing  one  of  cutting  up.  "Every  F  has  the  relation  R 
to  some  G"  into  the  components  "a  is  an  F"  and  "a  is  related 
by  R  to  no  G"  and  postulating  that  one  must  be  false.  In  the 
case  where  either  F  or  G  are  null,  it  is  the  first  component  which 
fails  of  acceptance. 

Frege  has  passed  over  some  logical  difficulties  here  which  need 
elucidation.  It  is  sufficiently  obvious  that  this  analysis  is  an 
afterthought  and  was  brought  in  by  main  strength  for  the  express 
purpose  of  dealing  with  Zero.  Moreover,  a  delicate  issue  is 
involved  here, — what  can  we  conclude  about  the  truth  or  false- 
hood of  a  proposition  whose  subject  does  not  exist?  Frege's 
method  is  to  chop  up  the  supposedly  complex  proposition  by 
analyzing  the  subject  into  its  logical  parts.  For  example, 
"The  green  horse  is  in  the  field"  becomes  equivalent  to  the 
falsehood  of  one  of  "The  horse  is  green"  and  "The  horse  is 
not  in  the  field."  As  a  consequence,  Frege  would  hold  that 
we  may  pasture  our  horse,  in  as  much  as  he  does  not  exist.  Now 
this  is  not  obvious  to  common  sense  and  needs  more  argument 
than  he  gives  it. 

Granted  the  definition  of  Zero,  Frege  finds  it  easy  to  define 
One  and  then  proceeds  to  discuss  the  law  of  the  natural  number 
series.  A  number  n  is  said  to  be  the  immediate  successor  of  m 
if  "there  is  a  concept  F  and  an  object  x  coming  under  it  such 
that  the  number  which  pertains  to  F  is  n  and  to  "objects  under 
F  not  identical  with  x,"  m.  It  is  important  to  observe  that  an 
assumption  has  been  made  here  without  explicit  mention.  This 
x  which  is  to  be  extracted  from  the  members  of  F  must  be  held 
fixed.  It  is  not  a  variable  but  must  be  so  definite  that  the  expres- 
sion "under  F  not  identical  with  x"  is  unambiguous.     In  other 


12  The  Notion  of  Number  and  the  Notion  of  Class 

words,  the  result  is  independent  of  what  choice  we  make  of  x, 
but  once  having  made  a  choice,  we  must  abide  by  it.  Now  this 
fact,  unless  we  admit  subjective  caprice,  involves  the  ability 
to  define  x  in  terms  of  known  properties  of  F.  It  does  not 
appear  obvious  that  this  is  always  possible. 

Upon  such  a  basis  Frege  claims  in  the  "Grundlage,"  his  early 
work,  to  be  able  to  establish  upon  the  basis  of  logic  alone,  the 
following  theorems : 

(i)  If  a  is  the  immediate  successor  of  Zero,  a  is  One. 

(2)  If  F  has  the  number  One,  then  a  is  an  F  is  not  always 
false. 

(3)  If  F  has  the  number  One,  a  is  an  F,  b  is  an  F,  then  a  and 
b  are  identical. 

(4)  The  relation  of  immediate  succession  is  one-one. 

(5)  Every  number  has  a  successor. 

From  our  present  standpoint  the  validity  of  (5)  may  well 
appear  doubtful.  It  insures  the  existence  of  just  as  many  con- 
cepts and  distinct  objects  as  there  are  numbers, — in  other  words 
infinitely  many.  If  (5)  is  derived  from  the  assumptions  of 
pure  logic  alone,  which  contain  no  ontological  or  existential 
elements,  its  truth  should  be  independent  of  how  many  distinct 
beings  there  are.  If  we  limit  our  widest  universe  of  discourse 
to  a  small  finite  number  of  terms  it  is  hard  to  see  how  the  exis- 
tence of  any  infinite  class  can  be  deduced  without  regarding 
terms  and  collections  of  terms  on  the  same  level,  and  this  is 
not  a  process  to  be  unhesitatingly  accepted. 

Husserl  in  his  "Philosophic  der  Arithmetik"  criticized  the 
"Grundlage"  and  one  of  the  points  he  makes  deserves  special 
mention.  He  recalls  Frege's  definition  by  abstraction  and 
points  out  that  this  method  does  not  define  the  content  of  the 
number  concept  but  its  Umfang.  Now  Umfang,  says  Husserl, 
if  it  is  to  mean  anything,  can  only  mean  the  collection  of  objects 
falling  under  thq  concept.  Thus  where  Frege's  definition  pur- 
poses to  make  number  a  definite  single  object,  it  really  makes 
it  a  plurality. 

Husserl  has  put  his  finger  upon  the  weakest  link  in  Frege's 
chain  of  argument.  Even  Russell,  admirer  of  the  definition 
as  he  was,  admitted  that  at  first  sight  it  must  appear  a  "wholly 
indefensible  paradox."  Clearly  the  definition  of  number  as  an 
"Umfang"  loses  all  claims  to  plausibility,  if  we  regard  this  to 


The  Notion  of  Number  13 

mean  a  mere  plurality  of  concepts,  and  like  Husserl  we  may  well 
wonder  how  any  one  could  have  thought  of  number  in  such  a 
strange  way.  However,  this  does  not  by  any  means  show  that 
Frege's  definition  is  as  faulty  as  Husserl  would  have  us  believe. 
It  is  a  convincing  proof  that  Frege  did  not  identify  the  "Um- 
fang"  of  a  concept  with  the  mere  enumeration  of  the  objects 
falling  under  it. 

Kerry  also  criticized  the  position  of  Frege  in  some  detail  but 
only  one  of  his  objections  has  any  point  to  it,  for  his  work  is 
based  upon  a  misunderstanding  of  the  aims  and  methods  of 
the  "Grundlage."  In  attempting  to  show  that  Frege  has  mis- 
takenly identified  concept  with  its  Umfang,  Kerry  remarks 
that  there  are  concepts  which  do  not  have  any  definite  Umfang 
at  all, — such  as  "heap"  which  is  clear  as  a  concept,  but  whose 
Umfang  is  obscure,  since  we  do  not  know  whether  we  are  going 
to  call  two  objects  a  heap  or  not.  This  amounts  to  a  demand 
for  an  analysis  of  the  process  of  definition. 

Let  us  enumerate  the  difficulties  we  have  found.  (1)  Given 
any  "Umfang,"  can  we  determine  upon  a  definite  member  of  it? 
(2)  When  is  a  concept  adequately  defined?  (3)  Can  the  exist- 
ence of  infinite  classes  be  deduced  from  pure  logic?  (4)  The 
notion  of  Umfang  is  so  indefinite  as  to  be  misunderstood  by  both 
Husserl  and  Kerry.  It  cannot  be  identified  with  a  mere  collec- 
tion from  the  extensional  point  of  view  or  Frege's  definition 
is  invalid.  What  is  the  logical  theory  of  an  Umfang,  then,  which 
will  support  the  definition?  (5)  Is  the  inclusion  of  the  null 
class  in  every  class  justified? 

Russell's  initial  discussion  of  number  occurs  in  the  eleventh 
chapter  of  the  "Principles."  He  lays  down  two  fundamental 
principles,  akin  to  Frege's, — (a)  numbers  are  to  be  regarded 
as  properties  of  classes,  (b)  "Two  classes  have  the  same  number 
when,  and  only  when,  there  is  a  one-one  relation  whose  domain 
includes  the  one  class,  and  which  is  such  that  the  class  of  corre- 
lates of  the  terms  of  the  one  class  is  identical  with  the  other 
class."  This  relation  of  equivalence  is  easily  shown  to  be 
reflexive,  symmetrical  and  transitive.  Russell  gives  his  reasons 
for  dissenting  from  Peano's  definition  which  postulates  the 
existence  of  a  common  property  whenever  we  have  a  field  of 
such  an  isoid  (reflexive,  symmetrical,  transitive)  relation. 

Peano  defines  number  as  this  common  property,  to  which 
the  equivalent  classes  have  an  identical  relation.     Russell  objects 


14  The  Notion  of  Number  and  the  Notion  of  Class 

to  definition  by  abstraction  in  general,  and  to  this  actual  defini- 
tion in  particular.  He  points  out  that  while  the  existence  of 
this  common  property  is  guaranteed  by  the  principle,  its  unique- 
ness is  not.  Two  ways  are  open,  the  definition  of  number  as 
the  whole  class  of  entities  to  which  all  equivalent  classes  have 
a  many-one  relation, — which  is  practically  useless,  and  the  other, 
"the"  number  of  a  class  as  the  class  of  classes  equivalent  to 
the  given  class.  This  last,  after  some  argument,  is  finally 
adopted.  Moreover,  Russell  announces  his  intention  of  hold- 
ing to  the  following:  "Whenever  Mathematics  derives  a  common 
property  from  a  reflexive,  symmetrical,  transitive  relation,  all 
mathematical  purposes  of  the  supposed  common  property  are 
completely  served  when  it  is  replaced  by  the  class  of  terms 
having  the  given  relation  to  a  given  term." 

The  tone  of  this  passage  is  noteworthy.  The  desirability 
of  a  definition  different  from  the  "class  of  classes"  is  not  denied; 
but  its  successful  accomplishment  is  despaired  of.  We  are  not 
able  to  define  number  as  a  common  property  because  of  defects 
in  the  logical  relations  of  predicates  and  classes  and  are  forced 
to  the  apparently  "indefensible  paradox"  as  the  only  way  out  of 
the  difficulty.  Now  why  should  Russell  see  paradox  where 
Frege  saw  none?  Clearly  Russell's  notion  of  a  class  is  more 
extensional  than  Frege's  and  he  sees  the  risk  of  making  one 
side  of  the  definition  (the  number)  singular  where  the  other  side 
(the  class)  is  plural.  Substituting  the  class  of  objects  having 
the  isoid  relation  for  the  desired  common  property  is  only  legiti- 
mate if  we  arc  given  a  clear  notion  of  what  is  meant  by  class, 
and  then  not  its  identification  with  a  mere  plurality  of  terms. 
Support  from  a  satisfactory  theory  of  classes  is  not  only  desir- 
able, but  absolutely  necessary. 

In  his  definition  of  multiplication,  in  purely  logical  terms, 
the  haste  with  which  the  "multiplicative  class"and  the  very 
delicate  issues  involved  in  the  assumption  of  its  existence  are 
passed  over,  is  striking.  In  the  "Principles"  no  doubt  seems 
to  be  felt  as  to  the  validity  of  the  process.  We  are  uncondi- 
tionally commanded  to  choose  and  to  keep  on  choosing.  Now 
here  is  not  only  Frege's  assumption  of  the  selecting  of  a  definite  x 
from  a  class  but  this  same  assumption  carried  to  an  infinite  power. 

Russell  bases  his  theory  of  the  natural  number  series  on  a 
set  of  postulates  and  indefinables  due  to  Peano.  The  indefin- 
ables  are  zero,  integer,  and  successor.     The  postulates: 


The  Notion  of  Number  15 

(1)  Zero  is  an  integer. 

(2)  If  a  is  an  integer,  the  successor  of  a  is  an  integer. 

(3)  If  a  is  an  integer  and  b  is  an  integer,  and  the  successor 
of  a  is  identical  with  the  successor  of  b,  then  a  and  b  are  identical. 

(4)  Zero  is  not  the  successor  of  an  integer. 

(5)  If  s  is  a  class  to  which  zero  and  the  successor  of  every 
integer  belonging  to  s  belong,  then  every  integer  belongs  to  s. 

Russell  correctly  remarks  that  another  assumption  is  neces- 
sary, i.  e., 

(6)  Every  integer  has  one  and  only  one  successor. 

It  is  very  important  that  this  should  be  explicitly  stated,  for 
it  formed  one  of  the  dubious  points  in  Frege's  theory. 

Russell  objects  to  Peano's  treatment  on  the  ground  that  it 
does  not  define  a  unique  system.  This  deficiency  is  to  be  rem- 
edied by  reducing  the  whole  to  purely  logical  terms.  Zero  is 
the  class  of  classes  whose  only  member  is  the  null  class.  Num- 
ber is  defined  as  before.  One  is  the  class  of  classes  not  null 
such  that  if  x  and  y  belong  to  the  class,  they  are  identical.  "  Hav- 
ing shown  that  if  two  classes  are  equivalent  and  a  class  of  one 
term  be  added  to  each,  the  sums  are  equivalent,"  the  successor 
of  n  is  defined  as  the  number  resulting  from  the  addition  of  a 
unit  to  a  class  of  n  terms.  As  a  result  Peano's  postulates 
are  satisfied  and  are  shown  to  possess  the  categoricity  and 
consistency  of  logic  itself.  "There  is,  therefore,  from  the  mathe- 
matical standpoint,  no  need  whatever  of  new  indefinables  or 
indemonstrables  in  the  whole  of  Arithmetic  or  Analysis." 

This  is,  of  course,  a  very  laudable  conclusion,  but  what  has 
become  of  our  added  postulate  (6)?  It  is  not  self-evident  that 
this  is  satisfied.  The  point  at  issue  is  the  logical  status  of  infinite 
classes.  Here  we  may  refer  to  a  later  passage  in  the  text  (p. 
357)  where  three  arguments  are  given  to  show  the  existence 
of  infinite  classes.  These  so-called  proofs  require  careful  scru- 
tiny. The  first  necessitates  the  gratuitous  assumption  that 
One  and  Being  are  distinct,  which  pure  logic  does  not  seem 
to  be  concerned  with.  The  third  argument  derives  its  plausi- 
bility from  a  strange  and  perverted  use  of  the  term  "idea." 
If  an  idea  is  not  a  psychological  or  physiological  reaction,  there 
is  no  reason  to  accept  the  premise,  "of  every  term  there  is  an 
idea."  Consequently  we  must  reject  both  of  these,  and  are 
thrown  back  upon  the  second  proof,  which  is  essentially  depen- 


1 6  The  Notion  of  Number  and  the  Notion  of  Class 

dent  upon  the  proposition  that  zero,  the  class  of  classes  equiva- 
lent to  the  null-class,  is  an  integer  which  has  a  successor  but 
not  a  predecessor.  Hence  so  far  as  the  "Principles"  is  con- 
cerned, Russell's  doctrine  of  number  stands  or  falls  with  the 
notion  of  zero. 

Here  we  have  our  old  difficulties  with  the  null-class,  and  we 
may  well  ask,  has  Russell  avoided  the  troubles  of  Frege?  In 
his  first  treatment  of  the  null-class  Russell  does  not  depart  from 
the  customary  attitude  of  symbolic  logic.  However,  he  shows, 
in  his  criticism  of  Peano,  that  all  is  not  such  smooth  sailing 
as  we  might  suspect  from  the  passage  referred  to  above.  The 
definition  of  the  null-class  as  included  in  every  class  is  objected 
to,  because  "there  are  no  such  terms  x;  and  there  is  a  grave 
logical  difficulty  in  trying  to  interpret  extcnsionally  a  class 
which  has  no  extension."  On  p.  68  we  find  that,  rejecting 
Peano's  identification  of  class  and  class  concept,  "there  is  no 
such  thing  as  the  null-class,  although  there  are  null-class  con- 
cepts." 

However,  a  new  problem  immediately  offers  itself.  Null 
class  concepts  possess  in  their  equality  an  isoid  relation,  conse- 
quently the  principle  of  abstraction  operates.  For  these  the 
required  common  term  to  which  they  have  the  same  relation 
must  be  taken  to  be  the  class  of  null  concepts.  "The  null-class, 
in  fact,  in  some  ways  is  analogous  to  an  irrational  in  Arithmetic; 
it  cannot  be  interpreted  on  the  same  principles  as  other  classes." 
Now  the  null  class  seems  to  be  not  a  fiction,  a  class  with  no 
terms,  but  a  class  of  infinitely  many  terms,  namely  all  null- 
concepts. 

Passing  on,  immediately  after  a  discussion  of  the  paradoxes 
we  meet  with  another  change.  Now  the  class  with  no  terms 
is  admitted,  inasmuch  "in  the  present  chapter  we  decided  that 
it  is  necessary  to  distinguish  a  single  term  from  the  class,  whose 
only  member  it  is."  Hegelian  dialectic  in  its  most  intricate 
stages  was  not  more  baffling  than  this.  At  one  moment  the 
null  class  has  no  members  and  is  rejected,  at  another  an  infinity 
of  members  and  is  accepted.  After  such  bewildering  changes 
we  should  feel  no  surprise  when  our  latest  report  shows  that 
it  is  again  without  members  and  accepted.  Worse  than  this, 
in  Appendix  B,  we  find  the  following:  "This  renders  the  very 
definition  of  Zero  erroneous;  for  every  type  of  range  will  have 


The  Notion  of  Number  17 

its  own  null-range,  which  will  be  a  member  of  Zero  considered 
as  a  range  of  ranges,  so  that  we  cannot  say  that  Zero  is  the  range, 
whose  only  member  is  the  null-range."  Finally,  in  the  Preface 
occurs  the  following  very  significant  remark:  "On  questions 
discussed  in  these  sections  I  discovered  errors  after  passing  the 
sheets  for  the  press;  these  errors,  of  which  the  chief  is  the  denial 
of  the  null  class  .  .  .  are  rectified  in  the  Appendices.  The 
subjects  treated  are  so  difficult  that  I  feel  little  confidence  in 
my  present  opinions." 

Where  the  propounder  of  the  theory  feels  little  confidence, 
we  may  not  be  blamed  for  feeling  less.  Russell,  however,  does 
not  seem  to  realize  than  an  irreproachable  definition  of  Zero  is 
indispensable  for  his  proof  that  Arithmetic  and  Analysis  are 
derived  from  logic.  Let  us  recall  the  difficulties  we  found  in 
the  work  of  Frege.  Selections,  infinite  classes,  the  null-class, 
have  all  been  discussed  in  the  "  Principles"  but  our  doubts  have 
not  been  dispelled.  On  the  problem  of  definition  Russell's 
position  may  be  imagined  from  his  remarks  on  formulas.  Can- 
tor and  Dedekind  thought  that  a  class  c  was  well  defined  if, 
given  x,  it  was  determinate  whether  x  belonged  to  c  or  not. 
Applied  to  the  theory  of  functions,  the  functional  relation  is 
only  specific  correspondence  and  any  particular  function,  to  be 
well-defined,  must  be  given  by  a  set  of  rules,  which  if  mutually 
independent,  must  be  finite  in  number. 

Russell's  view  is  quite  different.  "The  usual  meaning  of 
formula  in  mathematics  involves  another  element  which  may  also 
be  expressed  by  the  word  law.  It  is  difficult  to  say  precisely 
what  this  element  is,  but  it  seems  to  consist  in  a  certain  degree 
of  intensional  simplicity.  .  .  .  It  is  therefore  essential  to 
the  correlation  of  infinite  classes  should  be  one  in  which  .  .  . 
the  formula  should  be  one  which  we  can  discover.  I  am  unable 
to  give  an  account  of  this  condition,  and  I  suspect  it  of  being 
purely  psychological.  There  is,  however,  a  logical  connection.  . 
This  amounts  to  saying  that  the  defining  relation  of  a  function 
must  not  be  infinitely  complex.  .  .  .  This  condition,  though 
it  is  itself  logical,  has,  I  think,  only  psychological  necessity." 
Under  such  opinions  it  is  difficult  to  see  how  we  can  be  guaran- 
teed the  possibility  of  choosing  a  definite  member  out  of  an 
arbitrary  class,  whose  members  are  not  identifiable  as  individuals 
and  therefore  cannot  be  explicitly  distinguished.     Without  this 


1 8  The  Notion  of  Number  and  the  Notion  of  Class 

possibility  being  granted,  the  multiplicative  class  fails  to  exist 
for  even  a  finite  number  of  classes.  To  be  told  that  these  diffi- 
culties are  "purely  psychological"  is  poor  consolation. 

The  logical  foundations  of  Arithmetic  have  been  discovered  to 
rest  upon  the  notions  of  Zero,  Number,  and  Succession.  To 
each  of  this  trio,  so  far  as  the  "Principles"  goes  to  show,  dam- 
aging objections  may  be  raised,  and  to  sum  up  our  consideration 
of  Russell's  philosophy  of  arithmetic,  we  will  review  them  briefly. 

(a)  Zero  is  admitted  at  the  close  of  the  work  to  have  received 
no  proper  logical  interpretation.  The  doctrine  of  the  null- 
class,  although  symbolically  convenient,  is  not  as  yet  justified. 

(b)  Number  is  applicable  to  classes,  if  they  are  many.  If 
a  number  is  a  class,  it  must  be  one  or  the  definition  is  invalid. 
Hence  a  class  cannot  be  exclusively  many,  nor  exclusively  one, 
if  Russell's  definition  is  to  hold.  The  early  proposed  identifi- 
cation of  class  with  numerical  conjunction  invalidates  the 
definition  of  number. 

(c)  Succession  has  not  been  shown  to  be  always  possible 
unless  Zero  be  granted  a  correct  definition.  The  notion  of 
Predecessor  is  not  explained  without  a  theory  of  Selections  in 
some  way  contraverting  the  notion  of  formula  given  above. 

What  conclusions  are  we  to  draw  from  this  hasty  review? 
In  the  first  place,  it  is  clear  that  Russell  has  not  answered  the 
great  majority  of  our  objections  to  Frege  if,  indeed,  he  has 
answered  any  of  them.  For,  in  order  to  preserve  Mathematics 
as  a  derivative  of  Logic,  he  has  been  forced  to  depart  from  the 
conception  of  Logic  as  laid  down  early  in  the  book  and,  to  change 
this  science  almost  beyond  recognition  by  varying  the  use  of 
the  term  "class."  To  keep  Arithmetic,  therefore,  he  has  done 
violence  to  Logic. 

In  the  second  place,  if  we  survey  the  work  carefully,  we  see 
that  in  establishing  the  indefinite  extension  of  the  natural  num- 
ber series,  our  author  has  resorted  to  extremely  questionable 
means  and  we  may  doubt  whether,  after  all,  theoretical  arith- 
metic has  been  deduced.  Even  granted  the  changes  in  Logic, 
has  Arithmetic  been  gained?  We  are  forced  to  conclude  that 
in  this  case  Frege's  methods  have  not  been  bettered.  Moreover, 
by  making  numbers  classes  and  by  throwing  doubts  upon  the 
nature  of  classes  not  only  the  deduction  of  Arithmetic  but  its 
very  content  is  imperiled  and   the  foundation  of  mathematics 


The  Notion  of  Number  19 

is  shaken.     To  deduce  Arithmetic  and  save  Logic,  violence  has 
been  done  to  Arithmetic.     Both  sciences  are  common  sufferers. 

B.    The  Psychological  Aspect 

§1.     Introductory 

The  problem  of  number  is  now  to  be  considered  from  a  slightly 
different  point  of  view.  In  our  survey  of  the  philosophy  of 
arithmetic  we  have  seen  that  those  who  looked  upon  psychology 
as  a  science  based  entirely  and  fundamentally  upon  introspec- 
tion (Frege  must  be  included  here,  since  his  contempt  for  psy- 
chology is  due  to  his  belief  in  its  subjectivity),  have  disagreed 
radically  and  have  left  us  with  a  difficult  and  delicate  logical 
problem, — the  nature  of  classes.  The  question  now  naturally 
is  suggested, — would  the  result  have  been  entirely  different,  if 
another  method  had  been  used?  What  conclusions  might  be 
drawn,  if  observation  is  substituted  for  introspection? 

Such  an  investigation  has  to  do  with  the  observable  behavior 
of  men  toward  number,  and,  it  must  be  confessed,  is  beset  with 
as  yet  unsurmountable  technical  obstructions.  Psychology  as 
an  exact  science  is  still  in  its  infancy,  and  many  of  its  conclusions 
are  not  only  debated  by  skeptical  critics,  but  are  questionable 
in  themselves.  It  will  be  our  method,  therefore,  to  avoid  par- 
ticular assumptions  of  a  technical  character  and  to  confine  our- 
selves to  observations  of  a  general  nature,  not  confirmed,  perhaps, 
by  concrete  instances,  but,  at  any  rate,  not  definitely  refuted. 
In  brief,  we  shall  endeavor  to  state  not  a  solution,  but  a  hypo- 
thesis. 

The  customary  behavior  of  adults  is  so  complicated,  and  so 
entangled  with  extraneous  considerations  that  an  investigation 
does  not  promise  any  great  results.  What  we  shall  do,  therefore, 
is  to  confine  ourselves  to  the  more  elementar.y  types  of  behavior 
as  found  in  (a)  primitive  man,  (b)  children.  We  must  attempt 
to  find  whether  the  notion  of  number  is  a  particular  act,  a  mode 
of  behavior,  or  whether  it  is  not  reached  until  we  have  a  still 
higher  level  of  action.  If  the  consciousness  of  number  is  a 
definite  concrete  act,  we  shall  say  that  it  is  gained  by  an  indi- 
vidual quantitative  discrimination;  if  it  corresponds  to  a  type  of 
such  actions,  it  will  be  termed  a  percept,  and  if  it  is  not  attained 
in  any  of  these  ways  it  will  be  assumed  to  be  conceptual.     Thus 


20  The  Notion  of  Number  and  the  Notion  of  Class 

the  possible  conclusions  are  parallel  in  a  way  to  the  theories  of 
Mill.  Kant  and  Frege,  as  representative  of  the  three  great  classes 
of  the  philosophy  of  mathematics. 

§2.     The  Implications  of  Anthropology 

Just  how  primitive  man  perceived  number  is  a  matter  of  some 
doubt.  The  problem  has  been  considered  in  a  careful  way  by 
several  anthropologists,  among  them  Conant,  McGee,  Tylor  and 
Levy-IB ruhl.  It  is  necessary  to  take  into  account  the  scanty 
available  historical  material,  as  well  as  existing  stages  of  primi- 
tive culture. 

The  main  opportunities  in  this  field  are  afforded  by  the  savage 
number-words — since  these,  being  a  part  of  language,  indicate 
easily  observed  behavior.  It  might  be  imagined  that  a  con- 
sideration of  the  primitive  methods  of  counting  (as  by  fingers 
or  toes)  would  be  fundamental,  but  investigators  concur  in  the 
opinion  that  these  are  comparatively  late  developments.  As 
a  matter  of  fact  numerals  among  the  lower  types  of  savages  are 
disappointingly  few.  McGee  gives  examples  of  Brazilian  tribes, 
which  have  words  for  one,  two,  three,  and  four  and  another 
much  used  term  meaning  "many,"  "heap,"  or  "multitude." 
The  behavior  indicated  by  this  last  term  is  especially  significant 
since  in  effect  it  is  equivalent  to  "non-denumerable"  and  sug- 
gests a  way  of  attacking  a  quantitative  problem  where  the  limit 
of  number  has  already  been  reached. 

Conant  establishes  clearly  that  even  where  savages  were 
deficient  in  number-language,  they  were  not  deficient  in  the 
conception  of  number.  Serial  arrangements  give  rise  to  types 
of  act,  which  have  no  explicit  number  words  corresponding  to 
them.  In  fact,  almost  all  of  the  primitive  tribes  have  a 
definite  limit  to  their  counting,  which  corresponds  in  some 
measure  to  their  facility  in  their  perception  of  number. 

Two  outstanding  factors  in  their  attitude  toward  number 
seem  to  be  order  and  correspondence.  In  the  district  of  Adelaide, 
Australia,  Moorhous?,  the  explorer,  found  that  the  children  as 
they  come  into  the  world  receive  in  the  order  of  birth  numerical 
names,  as  follows: 

first  child,  if  boy  kertameru,  if  a  girl  kertanya 
second,  icarritya  warriarto 

third.  kudnutya  kudnarto 

fourth,  monaitva  monarto 


The  Notion  of  Number  21 

and  so  on,  up  to  nine.  Now  these  same  Australians  possess 
numerals  only  up  to  three,  so  we  can  imagine  the  savage  parent 
calling  the  roll  and  being  convinced  that  the  roster  of  his  children 
is  complete  without  having  the  necessary  knowledge  of  abstract 
number,  or,  it  may  be,  without  any  notion  whatever  of  abstract 
number.  He  does  not  count  up  to  nine  but  calls  names  up  to 
the  ninth.  He  has  a  comparatively  adequate  and  successful 
type  of  behavior  toward  a  particular  group, — his  children,  but 
his  numerical  acts  in  other  directions  are  complete  failures. 
It  appears,  therefore,  that  number  is  not  gained  by  the  perception 
of  order,  or  by  mere  perception  of  any  kind. 

The  Murray  Islands  in  Torres  Straits  offer  an  interesting 
example  of  the  further  development  of  number-perception  by 
means  of  correspondence,  and  show  that  this  process  of  tallying 
off  is  of  fundamental  importance  psychologically.  The  natives 
of  these  regions  have  only  two  numbers,  netat  (one)  and  neis 
(two).  Beyond  this,  they  proceed  by  repetition,  such  as  neis  neis 
for  four,  or,  more  generally  by  correspondence, — a  fixed  method 
of  reference  to  the  parts  of  the  body.  By  this  last  device  they 
can  count  up  to  thirty-one.  They  commence  with  the  little 
finger  of  the  left  hand  and  go  from  the  fingers  to  the  arms, 
shoulders,  etc.,  ending  with  the  little  finger  of  the  right  hand. 
If  these  are  not  sufficient,  the  islander  has  recourse  to  the  toes 
and  the  lower  part  of  the  anatomy.  The  number  of  any  desired 
group  is  indicated  by  the  requisite  part  of  the  body,  and  then 
starting  from  the  little  finger  of  the  left  hand,  the  embryo  expon- 
ent of  one-one  correspondence  ascertains  the  relative  position 
of  the  particular  part  desired.  In  such  behavior  as  this  we 
have  the  extension  of  successful  behavior  toward  a  particular 
group,  namely  the  parts  of  the  human  organism  over  into  any 
other  arbitrary  field  by  reference  to  this  given  particular  group. 
Even  on  this  higher  level,  however,  we  have  not  yet  found  the 
abstract  numbers  of  arithmetic. 

The  transition  to  the  formation  of  a  number  system  may  be 
made  in  many  ways,  a  discussion  of  which  would  be  interesting 
but  hardly  important  for  our  purposes.  Whatever  base  be 
chosen,  be  it  two,  five,  ten  or  what  not,  the  fact  that  there  is  a 
base  is  for  us  the  fundamental  issue.  It  gives  us,  so  to  speak, 
the  time  for  our  numerical  melody,  the  rhythm  of  the  scale 
helping  us  to  remember  and   to  group  while  counting.     The 


22  The  Notion  of  Number  and  the  Notion  of  Class 

distinction  between  the  possessors  of  a  system  based  on  the 
fingers  and  such  more  primitive  numberers  as  the  Murray 
Islanders  is  well  illustrated  by  the  Baikaris.  A  member  of  this 
tribe  touches  the  counted  objects  with  the  right  hand,  holding 
up  corresponding  fingers  of  the  left  as  he  does  so,  then  for  groups 
above  five  he  reverses  the  process.  He  does  not  count  the 
fingers  themselves  but  counts  with  the  fingers  and  may  continue 
enlarging  this  process. 

We  have  seen  that  the  different  tribes  have  varying  limits 
to  their  counting  which  correspond  to  their  mental  rank.  Be- 
sides the  gradual  extension  of  this  limit  to  include  larger  groups, 
an  extension  of  numbers  in  generality  is  necessary.  For  instance, 
a  Canadian  tribe  has  the  following  words  for  groups  of  two 
objects  depending  on  the  qualitative  character  of  the  objects: 
Maalck,  Masem,  Mats'ak,  Matlgsa,  Matlouth,  Matsemala, 
Maalis,  etc.  The  extraction  of  the  common  root  is  the  indis- 
pensable prerequisite  to  the  gaining  of  the  number  concept. 

The  number-perception  of  the  savage  may  far  outrun  his 
words  to  describe  it  and  he  may  have  an  intuitive  sense  of  num- 
ber which  defies  clear  perception  and  which  demands  for  its 
expression  the  words  for  "multitude"  or  "heap."  This  sig- 
nifies not  only  his  inability  to  apply  a  numeral  to  the  presented 
group,  or  his  failure  to  refer  it  successfully  to  the  parts  of  the 
body,  but  his  lack  of  conviction  and  assurance.  There  seems 
to  be  a  fundamental  factor  in  human  nature  which  demands 
a  vague  and  doubting  attitude  toward  that  which  cannot  be 
counted  or  numbered.  We  do  not  mean  to  apply  the  intro- 
spective method  here  by  the  use  of  such  terms  as  "conviction" 
and  "assurance."  They  are  to  be  defined  in  terms  of  the  time 
elapsed  between  the  presenting  of  a  situation  and  the  action 
which  aims  at  the  successful  handling  of  that  situation. 

Any  attempt  to  go  back  of  the  actually  observed  data  of  the 
anthropologists  is,  of  course,  pure  speculation.  Speculation, 
however,  although  hazardous,  is  not  always  unprofitable  and 
the  result  may  justify  the  means.  Can  we  not  conclude,  with 
some  measure  of  plausibility,  that  before  explicit  counting,  two 
species  of  number-behavior  existed; — a  clear  cut,  swift  and 
assured  reaction  to  a  presented  group,  and  a  type  of  act  some- 
what similar  to  the  first  but  lacking  in  precision  and  decision, 
doubtless  accompanied  by  doubt  and  uncertainty?     The  pro- 


The  Notion  of  Number  23 

gressive  development  of  number  and  general  number  words 
represents  from  this  point  of  view  a  gradual  and  slow,  but  none 
the  less  persistent  conquest  of  the  residuum  of  hesitation  and 
ignorance.  Might  we  not  indeed  go  further  and  trace  the  origin 
of  number  to  this  very  type  of  doubtful  act?  First  we  might 
have,  let  us  say,  a  specific  act  of  discrimination  such  as  the  crow 
performs  when  it  is  able  to  behave  differently  when  one  of  a 
group  of  four  is  absent,  or  like  the  nightingale's  selection  of 
three  worms.  This  is  repeated  until  a  type  of  successful  dis- 
crimination is  present  which  by  continual  practice  becomes 
habitual  under  a  certain  situation.  Here  we  have  swift,  con- 
fident action. 

If  we  concede  to  man,  who  is  defined  as  the  possessor  of  free, 
rational  behavior  the  ability  not  to  do  different  things  at  the  same 
time  but  the  ability  to  do  the  same  thing  under  different  circum- 
stances, we  yield  to  him  the  power  of  extending  a  type  of  success- 
ful behavior  into  other  fields.  Granted  this,  which  is  the  very 
root  and  foundation  of  abstract  thought,  we  can  clearly  see 
how  elementary  particular  quantitative  discriminations,  the 
mere  perception  of  change  or  invariance  in  magnitude,  leads 
by  its  very  vagueness  to  an  attempt  to  differentiate  explicitly 
and  to  act  in  conformity  with  some  established  custom  or  habit. 
From  the  mere  perception  of  such  change  or  invariance,  which 
is  undoubtedly  possessed  in  some  degree  by  the  lower  animals, 
we  pass  to  the  recognition  of  different  changes,  and  from  that 
to  a  type  of  constant  behavior  toward  varying  groups  of  the 
same  changes.  Under  this  drying  out  process  of  extension 
and  abstraction,  qualitative  differences  gradually  evaporate, 
until,  at  last,  when  abstract  number  is  reached,  the  complexity 
of  the  preceding  process  is  elided  and  forgotten. 

The  number  words  do  not  evolve  from  this  struggle  for  mastery 
of  quantitative  change,  but  seem  to  traverse  parallel  lines.  In 
their  rudimentary  forms,  they  are  bestowed  with  all  manner 
of  religious  and  imaginative  significance  and  are  not  applied 
abstractly  to  presented  groups.  McGee  says,1  "These  number 
systems  are  distinct  from  Aryan  arithmetic.  .  .  .  They  are 
devices  for  binding  the  real  world  to  the  supernal,  and  it  is  only 
in  an  ancillary  way  that  they  are  prostituted  to  practical  uses 
.     .     .     yet  by  reason  of  the  extraordinary  potency  imputed 

1  American  Anthropologist,  1899. 


24  The  Notion  of  Number  and  the  Notion  of  Class 

to  them  they  dominate  thought  and  action  in  the  culture  stages 
to  which  they  belong." 

The  actual  employment  of  numerals  seems  to  be  the  meeting 
point  of  two  tendencies,  therefore, — (i)  A  demand  from  prac- 
tical sources  for  a  universal  term  applicable  to  the  similarity 
observed  among  groups  of  qualitatively  different  objects.  (2) 
An  emotional  and  personal  reaction  toward  an  ordered  series, 
in  which  rhythm  plays  an  important  part — any  one  who  has 
heard  children  count  while  skipping  rope  can  appreciate  this. 
From  this  sense  of  rhythm  we  can  see  the  development  of  the 
systems  of  numeration.  The  binary  scale  comes  first;  with 
more  fully  developed  mentality  we  have  the  other  bases.  The 
decimal  scale  is  a  surprisingly  late  development,  showing  clearly 
that  the  finger- toe  method  is  not  fundamental  and  that  number 
is  not  on  the  perceptual  level.  The  so-called  mystic  properties 
of  numbers  are  probably  derived  from  the  affective  reaction 
toward  the  ordered  scale,  the  numbers  seven  and  thirteen  being 
especially  interesting  in  this  respect.  Such  emotional  factors 
are  markedly  present  in  the  beliefs  of  the  Pythagoreans,  where 
they  ascribed  mystic  virtues  to  all  the  simple  numbers.  It  is 
clear  that  no  mere  practical  demand  leads  to  the  construction 
of  a  mathematical  cosmology;  a  certain  esthetic  craving  must 
be  super-added. 

We  seem  to  be  led  to  the  conclusion  that  primitively  number 
is  both  logical  and  esthetic — not  only  a  universal,  but  a  uni- 
versal with  an  emotional  value  of  varying  degree,  that1  "  (1) 
the  origin  of  number  names  is  at  the  bottom  of  the  scale  of 
human  development;  (2)  primeval  man  does  not  cognize  decimal 
systems;  (3)  does  not  use  his  fingers  and  toes  as  mechanical 
adjuncts  to  nascent  notation." 

With  the  passing  of  time  and  the  banishment  of  animism 
the  mystic  powers  of  numbers  tend  to  become  disregarded  and 
number  becomes  more  and  more  an  abstract  means  of  adjustment. 
The  fingers  and  toes  appear  as  practical  aids;  the  abacus,  marks 
in  the  sand,  large  and  small  pebbles,  etc.,  are  used.  Order 
and  rhythm  on  the  one  hand,  correspondence  on  the  other  make 
these  modes  of  behavior  habitual  and  successful.  Finally  the 
concrete  aids  to  tallying  become  replaced  by  symbols  and  reck- 
oning develops  into  arithmetic  and  that  into  algebra.  If  this 
is  correct,  the  numbers  of  pure  mathematics  must  be  not  only 

1  McGee,  op.  cit. 


The  Notion  of  Number  25 

universals,  but  concepts  of  the  very  highest  generality  and  order 
of  abstraction, — and  anthropology  fails  to  confirm  the  psy- 
chological views  of  Mill  and  Kant. 

§3.     The  Results  of  Experiment 

There  is  a  striking  parallel  between  the  child  and  the  primitive 
man  in  the  ability  of  each  to  grasp  concrete  number  before 
acquiring  a  knowledge  of  numerals,  or  even  the  ability  to  count. 
Children  can  reproduce  a  group  numerically  even  when  giving 
it  the  wrong  numeral,  after  learning  the  names;  for  the  associa- 
tion of  the  mechanically  learned  series  of  names  with  the  def- 
initely given  group  may  be  made  erroneously  or  not  at  all. 
Phillips1  cites  a  case  where  a  child  who  knew  the  words  for  num- 
bers and  was  able  to  recite  them  in  order,  on  counting  sticks 
was  as  apt  to  call  the  third  one  "six"  as  "three."  There  seems 
to  be  a  strong  element  of  rhythm  in  the  motor  activities  of 
counting,  which  is  not  always  successfully  transferred. 

It  is  not  until  the  process  becomes  mechanical  by  frequent 
repetition  that  the  names  are  applied  correctly  and  the  connec- 
tion between  name  and  group  is  perceived.  However,  the 
motor  tendencies  are  still  apparent.  The  child  is  likely  to  count 
in  sing-song  and  will  be  much  more  accurate  if  allowed  to  observe 
a  definite  rhythm.  All  experience  agrees  that  counting  of  con- 
crete groups  must  become  familiar  and  habitual,  before  the 
abstract  character  of  number  is  understood.  It  is  the  accepted 
procedure  in  elementary  arithmetic  to  introduce  many  concrete 
problems  such  as  "3  apples  and  5  apples  make  how  many  ap- 
ples?" This  is  not  without  significance.  It  shows  that  the 
gradual  transition  of  number  from  a  perceptional  reaction  cannot 
be  eliminated.  The  too  hasty  progress  of  our  common  schools 
through  the  mechanical  routine  of  methodological  devices  fre- 
quently leads  to  a  surprising  knowledge  of  mathematical  machin- 
ery with  an  equally  surprising  ignorance  of  what  it  is  all  about. 
This  can  be  directly  traced  to  the  abrupt  transition  from  names 
of  numbers  to  a  parrot-like  memorization  of  rules  for  their 
manipulation  without  enough  concrete  illustrations  being  given 
to  induce  any  realization  of  the  meaning  of  these  rules.  Having 
once  started  taking  rules  for  juggling  symbols  on  faith,  the 
student  is  prepared  for  the  easy  descent  to  that  mathematical 
Avernus,  of  which  the  type  of  pupil  who  can  solve  any  literal 

1 "  Number  and  its  Applications,"  Ped.  Sem.,  Vol.  V,  p.  262. 


26  The  Notion  of  Number  and  the  Notion  of  Class 

problem  in  Algebra,  while  helpless  before  one  couched  in  con- 
crete terms,  is  an  all  too  familiar  denizen.  No  matter  how 
abstract  and  formal  a  science  Mathematics  has  become,  the 
way  to  its  comprehension  is  by  the  gradual  ascent  from  percept 
to  concept,  from  act  to  type  of  act. 

The  perception  of  number  has  been  carefully  studied  by  a 
number  of  experimenters,  notably  Messenger,  Burnett,  Dietze, 
Arnett,  Nanu,  Lay,  and  Howell. 

It  is  important  from  a  psychological  as  well  as  a  philosophical 
point  of  view  to  decide  whether  the  apprehension  of  number  is 
gained  by  counting  (or  some  equivalent  process),  or  is  due  to 
an  immediate  perception,  and  in  addition,  to  determine  how 
far  accurate  numbering  can  be  carried  without  actual  counting. 

Cattell  found  that,  where  certain  lines  were  exposed,  up  to 
four  or  five  the  estimation  was  correct  and  that  in  case  of 
failure  the  number  was  likely  to  be  underestimated.  Dietze 
experimented  with  metronome  beats,  and  discovered  that  correct 
judgment  depended  on  the  rapidity  of  the  beats  and  their 
rhythmical  grouping.  On  the  most  favorable  conditions  eight 
groups  of  five,  forty  beats  in  all,  were  accurately  perceived. 
How  far  Dietze  succeeded  in  eliminating  counting  is  dubious. 
Warren  used  the  reaction  method  and  had  his  subjects  open 
their  mouths  when  they  apprehended  the  group  (in  this  case 
one  to  eight  circles).  Three  simultaneous  objects  and  five 
presented  successively  were  correctly  estimated. 

Nanu  repeated  the  experiment  of  Dietze's,  in  a  more  careful 
way.  The  indicated  results  were  that  in  every  case  the  limit 
reached  as  high  as  eleven  beats  and  in  some  instances  up  to 
forty-nine.  All  the  subjects  involuntarily  subjected  the  beats 
to  a  kind  of  rhythm.  When  sets  of  points  were  presented,  the 
greatest  number  of  simultaneously  given  points  was  five,  of  suc- 
cessive, six.  The  experiments  also  showed  that  linear  arrange- 
ments were  not  as  favorable  as  symmetric  and  that  circular  and 
polygonal  arrangements  are  undesirable.  The  number,  more- 
over, of  simultaneously  given  elements  successfully  numbered 
is  greater  than  Warren  thought,  being  eight  to  ten  under  the 
best  conditions.     Counting  was  eliminated  as  far  as  possible. 

An  exceedingly  valuable  account  of  these  researches  has  been 
given  by  Howell  in  his  "Foundational  Study  in  the  Pedagogy 
of  Arithmetic,"  to  whom  the  following  observations  are  in  large 


The  Notion  of  Number  27 

part  due.  It  seems  to  be  the  consensus  of  opinion  among  investi- 
gators that  the  apprehension  of  number  varies  with  the  way  the 
group  is  presented,  a  theory  which  has  enough  superficial  plausi- 
bility about  it  to  render  it  acceptable.  For  instance,  Messenger 
found  that  larger  objects  in  a  given  space  give  the  idea  of  a  greater 
number;  that  the  perception  of  number  is  influenced  partly 
by  spatial  qualities  and  partly  by  other  considerations.  In  the 
majority  of  cases  the  perception  is  due  to  the  association  of 
qualitative  differences  in  the  unit  with  numbers  of  parts  derived 
from  actual  count.  With  the  establishment  of  the  association 
it  becomes  mechanical  and  number-perception  seems  immediate. 
The  apprehension  of  large  groups  is  due  to  either  rapid  mechani- 
cal counting  or  the  association  of  number  and  form. 

Arnett's  experiments  with  counting  showed:  "Any  lack  of 
rhythm  caused  the  count  to  be  very  difficult  and  inaccurate. 
.  .  .  .  Irregular  counting  may  be  made  easier  and  more 
accurate  by  practice."  Here  we  have  again  the  intimate  con- 
nection of  rhythm  and  the  number  scale.  The  actual  corres- 
pondence between  objects  and  numerals  was  found  to  necessitate 
some  motor  reaction,  as  touching  the  objects,  nodding  the  head, 
etc.  From  the  mere  citing  of  names  as  a  motor  response,  to  the 
mechanical  tagging  of  objects  with  these  names,  the  child  passes 
to  a  realization  of  the  immense  range  of  application  of  the  method 
and  the  subsequent  abstraction. 

Lay  opposed  the  experimenters  such  as  Messenger,  who  held 
counting  to  be  implicitly  present  in  numerical  judgments.  His 
work  aims  at  showing  that  an  immediate  apprehension  of  number 
is  possible.  He  made  a  laborious  and  careful  series  of  experi- 
ments with  school  children  in  which  various  types  of  number- 
pictures  were  used  as  stimuli,  and  came  to  the  following  con- 
clusions: 

(1)  The  faculty  of  number-perception  varies  with  (a)  the 
power  of  clear  imagery,  (b)  the  homogeneity,  distance,  size, 
and  color  of  the  objects. 

(2)  It  is  improved  when  the  sense  of  touch  is  used  along  with 
the  sense  of  sight.  The  number  image  unites  objects  spatially 
and  temporally. 

(3)  The  particular  number-perception  is  independent  of  any 
arrangement  in  a  spatio-temporal  series.  The  row  form  is  not 
necessary  for  the  apprehension  of  number. 


28  The  Notion  of  Number  and  the  Notion  of  Class 

(4)  The  images  of  the  elementary  numbers  are  immediately 
perceived;  those  of  the  higher  numbers  (above  ten)  are  gained 
by  the  visualization  of  the  elementary  cardinal  numbers  in 
connection  with  collections  of  tens. 

(5)  Those  who  hold  that  perception  of  number  depends  upon 
counting  have  no  valid  psychological  basis  for  their  argument. 

Such  a  position  as  Lay's  is  extremely  radical.  He  over- 
emphasizes perception  almost  as  much  as  his  opponents  over- 
rate counting.  His  conclusions  have  been  challenged  and 
various  attempts  made  to  check  them  up  by  parallel  series  of 
experiments.  Of  these  the  most  careful  have  been  those  of 
Howell.  He  comes  to  the  conclusion  that  Lay's  theories  are 
not  corroborated,  although  experimental  results  tend  to  cast 
increasing  doubt  on  the  theory  that  number  is  derived  purely 
and  simply  from  the  process  of  counting.  He  arrives  at  one 
illuminating  result:  "The  number  pictures  themselves  consti- 
tute for  the  child  a  tremendous  step  toward  complete  abstrac- 
tion, in  that  he  is  taken  away  from  the  specific  numbered  objects 
of  various  kinds  to  a  representation  that  may  stand  for  any  of 
them."  The  discovery  of  Lay  that  images  of  successions  lead 
to  particularity  much  more  than  images  of  groups  is  substanti- 
ated. It  is  also  found  that  number  concepts  arise  rather  late 
in  children,  and  Howell  recommends  that  the  introduction  of 
abstract  number  be  postponed  until  the  third  or  fourth  year  of 
school. 

Counting  is  found  to  play  the  largest  part  in  numbering 
among  those  of  an  auditory  type,  and  for  this  reason  it  is  unjust 
to  neglect  the  process,  as  Lay  does.  His  view  that  the  percep- 
tion of  a  group  "becomes,  through  a  Kantian  reaction,  numerical 
apprehension"  is  not  confirmed,  although  it  is  not  proved  that 
counting  is  an  indispensable  prerequisite.  The  exact  deter- 
mination of  this  issue  is  a  matter  of  great  delicacy,  and,  pending 
its  unlikely  settlement,  an  attitude  of  compromise  is  necessary. 
"The  instantaneous  grasp  of  a  group  of  objects  visually  pre- 
sented is  not  intuitive.  .  .  .  It  is  made  possible  by  repeated 
prior  experiences  of  association  of  certain  qualities  differing 
in  the  unit  expression.  .  .  .  Lay's  notion  that  the  appre- 
hension is  immediate  cannot  be  sustained.  .  .  .  The  burden 
is  on  him  to  show  that  these  children  did  not  practise  a  primitive 
method  of  numeration." 


The  Notion  of  Number  29 

Howell  is  certainly  right  in  rejecting  the  hasty  conclusion 
that  the  immediate  perception  of  number  is  possible.  Proof 
of  such  a  statement  involves  not  only  great  logical  difficulties, 
but  must  also  surmount  formidable  technical  obstructions. 
Moreover,  on  the  face  of  it,  it  is  unlikely  that  any  abstraction 
of  the  nature  of  pure  number  should  be  intuitively  perceived 
by  a  child.  Such  a  faculty  would  involve  a  discontinuity  in 
intellectual  development  which  would  be  hard  indeed  to  explain. 
Transition  from  the  repeated  concrete  to  the  abstract  is  the 
only  power  we  have  any  right  to  postulate  in  the  child.  That  a 
sharp  break  in  the  process  occurs  in  the  special  case  of  number 
is  an  assumption  to  be  made  only  in  the  case  of  dire  necessity, 
and  such  necessity  by  no  means  exists. 

The  exponents  of  counting  are  not,  however,  to  be  completely 
agreed  with.  They  claim  that  the  visual  apprehension  of  a 
group  numerically  contradicts  the  psychological  principle  that 
the  field  of  consciousness  is  limited,  that  only  a  relatively  small 
number  of  objects  can  be  grasped  simultaneously.  This  view, 
however,  neglects  the  power  of  grouping,  by  which  the  visualiza- 
tion of  the  number-pictures  is  widely  extended  and  which  is 
greatly  facilitated  by  practice.  Even  after  a  group  is  counted, 
the  simultaneous  grasp  of  it  soon  follows,  and  this  can  be  so 
quickened  by  practice  that  counting  becomes  unnecessary.  The 
image  replaces  the  serial  count. 

Another  charge  of  the  advocates  of  counting  is  that  children 
who  would  be  taught  to  visualize  number  by  grouping  instead 
of  counting  would  be  restricted  to  their  images  and  would  never 
attain  pure  number  in  its  abstraction.  Although  it  is  no  doubt 
true  that  counting  makes  this  abstraction  easier,  that  does  not 
show  by  any  means  that  it  could  not  take  place  without  it.  More- 
over, the  visualized  images  aid  in  calculation  as  a  check  on  the 
work.  Speaking  generally,  it  is  quite  easy  to  gain  a  false  mathe- 
matical result  by  plausible  symbolical  consideration,  where 
concrete  illustrations  (say  by  a  geometrical  figure),  if  introduced, 
provide  less  rapid,  but  far  safer  means  of  progress.  The  child 
who  has  distinct  recollection  of  five  as  applied  to  all  the  fingers 
of  one  hand  and  six  as  adding  one  finger  of  the  other  hand  is 
far  less  likely  to  say  one,  two,  three,  four,  six,  than  the  child 
who  is  merely  learning  words  by  rote. 


30  The  Notion  of  Number  and  the  Notion  of  Class 

Thus  we  see  that  there  is  unquestionably  ground  for  the  asser- 
tion that  number  is  not  essentially  dependent  upon  order  and 
counting  although  there  is  no  reason  to  go  to  the  extremes  of 
Lay.  Children  manifestly  possess  the  ability  to  make  quanti- 
tative differentiations  independent  of  order,  this  ability  varying 
with  their  ability  to  visualize  and  the  color,  shape,  etc.,  of  the 
objects.  This  ability  may  be  developed  until  accurate  observa- 
tions of  absolute  number  are  made  with  confidence.  A  bank 
cashier,  for  example,  does  not  need  to  count  a  small  number  of 
coins  lying  before  him;  he  groups  them  and  behaves  just  as  if 
he  had  counted. 

The  experimental  results  show,  therefore,  a  striking  parallel 
between  the  attitude  toward  number  of  the  child  and  the  behav- 
ior of  primitive  man.  There  is  the  same  elementary  quantitative 
distinction  developing  by  order  and  correspondence  through  prac- 
tical application  to  the  pure  concept;  there  is  also  the  esthetic 
factor  of  rhythm  which  makes  for  the  homogeneity  of  the  number 
scale.  The  numbers  of  theoretical  arithmetic  are  show  in  attain- 
ment and  abstractly  conceptual  in  character. 

§4.     The  Philosophers  Again 

We  may  now  resume  our  discussion  of  the  metaphysical  the- 
ories, this  time,  with  a  view  as  to  their  psychological  implica- 
tions, looked  at  abstractly.  The  problem  we  have  now  before 
us  is:  What  is  the  nature  of  the  experience  which  leads  to  the  notion 
of  abstract  number?  Hoiv  is  it  that  man  is  capable  of  mathematical 
behavior?  We  may  neglect  the  crudities  of  Mill  and  the  em- 
piricists who  would  reduce  mathematics  to  the  level  of  quan- 
titative discrimination.  If  our  genetic  investigations  have  told 
us  anything,  they  have  taught  us  that  the  notion  of  number  is 
far  above  the  level  of  immediacy. 

It  seems  to  be  an  implicit  presupposition  among  philosophers 
that  whatever  is  to  be  counted  must  first  be  collected;  and,  with 
this  in  mind,  we  can  see  plainly  that  the  differences  between 
the  various  theories  are  due  to  disagreement  about  the  psy- 
chological character  of  collecting.  Individual  elements  com- 
pared as  such  do  not  give  either  the  notion  of  plurality  or  of 
definite  number.  To  the  elements  must  be  added  the  property 
of  their  being  a  whole  and  the  question  which  is  thus  raised  is, — 


The  Notion  of  Number  31 

what  is  this  collective  principle?  What  is  the  psychological 
basis  for  our  combining  elements? 

A  possible  answer  which  might  be  advanced  is  that  the  required 
common  property  is  simply  co-presence  in  consciousness  which 
is,  abstractly  considered,  equivalent  to  having  a  common  relation 
to  a  single  observer.  To  this  we  may  reply  that  many  things 
are  present  in  consciousness,  which  we  do  not  treat  as  collections. 
The  combining  process  is  essentially  teleological ;  to  unite  the 
many  into  a  single  whole  our  interest  must  be  elicited  and  our 
attention  thus  far  focused.  If  the  theory  we  are  considering 
were  tenable,  in  our  experience  at  a  given  moment  there  would 
be  the  material  for  one  and  only  one  collection;  while  on  the 
contrary,  it  is  a  familiar  fact  that  we  have  the  material  for  many 
concepts  constantly  before  us. 

There  is  a  latent  confusion  here  which  it  would  be  well  to 
bring  to  light,  since  in  the  sequel  we  shall  find  ourselves  frequently 
beset  by  it.  The  distinction  which  has  been  omitted  is  an 
essential  one.  In  no  distinction  of  this  kind  can  it  be  safely  over- 
looked that  there  is  a  difference  between  a  subject  behaving  toward 
an  object,  experiencing  it;  and  that  same  subject  reflecting  on  his 
experience.  The  relation  in  the  first  case  is  comparatively 
simple;  from  the  reflective  standpoint  we  have  a  set  of  relations 
which  are  mediated  and  complex.  When  we  hastily  write 
down  subject-relatcd-to-objects  it  seems  as  a  direct  consequence 
of  strict  logic  that  a  collection  of  objects-&«0w«-&y-subject  must 
exist,  which  would  of  course  prove  what  is  required.  There  is 
no  trouble  in  the  logic;  it  is  with  the  premise  that  we  must 
quarrel.  We  have  no  right  to  assume  subjcct-knoiving-objects, 
for  by  so  doing  we  are  putting  ourselves  in  the  reflective  stand- 
point and  any  conclusion  we  arrive  at  exists  only  for  its  reviewing 
the  process.  The  subject  knows  the  objects  as  such  and  not 
as  known.  Consequently  the  common  quality  of  being  known 
they  do  not  possess  for  him. 

The  temporal  theory  is  not  essentially  different  from  this. 
We  may  dismiss  rather  summarily  the  notion  that  to  collect 
elements,  they  must  be  simultaneously  presented.  To  grasp 
them  as  a  collection,  we  must  in  some  fashion  know  them  at  the 
same  time;  but  this  by  no  means  shows  that  we  know  them  to  be 
temporally  coexistent.  The  necessary  similarity  is  hardly  the 
possession  of  the  same  time  abscissa. 


32  The  Notion  of  Number  and  the  Notion  of  Class 

The  view  of  Kant  is  more  subtle.  Instead  of  a  cross-section 
of  time  a  longitudinal  view  is  taken.  The  contention  is  that 
succession  is  the  important  factor,  thus  placing  the  required 
similarity  among  the  mutual  time  relations  of  the  objects.  In 
defense  of  this  view  it  is  alleged  that  we  can  attend  to  but  one 
thing  at  a  time.  Granted  this,  which  is  at  best  extremely 
dubious,  we  cannot  concede  that  the  situation  of  the  temporal 
theory  is  a  whit  more  promising.  If  we  attend  to  a  succession 
and  know  it  to  be  a  succession  we  recall  all  the  past  objects  and 
in  a  given  instant  have  the  whole  collection  present  before  us. 
We  do,  of  course,  know  and  attend  to  the  group  as  one  but  we 
just  as  much  attend  to  it  as  many;  and  the  whole  positive  force 
of  the  argument  we  have  been  considering  lies  in  the  tacit  elision 
of  half  the  process  and  that  half  wherein,  as  a  matter  of  fact, 
the  crux  of  the  difficulty  lies.  It  may  very  well  be  questioned 
whether  temporal  conditions  are  even  a  necessary  basis  for  the 
required  common  property.  To  be  sure,  the  events  we  do 
unite  have  a  temporal  beginning  and  passing  away,  but  we 
cannot  be  sure  that  what  is  ever  present  is  never  irrelevant. 
Here  again  we  must  emphasize  the  difference  in  standpoints; 
that  while  for  us  reviewing  the  process  the  time  relations  stand 
out  clear  and  distinct,  for  the  actual  experient  they  were  slurred 
over  and  deduced  later,  giving  a  logical  and  not  a  psychological 
condition. 

The  spatial  theory  is  no  better.  To  Schuppe's  ingenious 
theory  that  the  counting  of  non-spatial  objects  is  only  possible, 
if  we  locate  them  in  imaginary  space,  we  may  reply:  (a)  any 
particular  spatial  relations  are  quite  irrelevant.  Two  men  are 
two  men,  whether  walking  arm  in  arm  on  Broadway  or  separated 
by  the  width  of  the  Pacific;  (b)  if  it  is  asserted  that  spatial 
relations  in  general  are  necessary — that  is,  that  the  objects  must 
possess  some  spatial  relations, — to  this  we  may  reply  that  this 
fact  si  seen,  if  at  all,  only  after  reflection,  the  object  of  immediate 
perception  being  the  particular  space  relation.  Thus  in  either 
case  the  theory  falls. 

The  opinion  of  Jevons  that  number  is  the  form  of  abstract 
difference  betrays  a  lack  of  analysis  of  the  notion  of  differentia- 
tion. The  psychological  process  of  conscious  distinctive  is  a 
type  of  act  resulting  from  repeated  comparisons.  Difference 
is  a  concept  which  derives  its  force  from  denying  a  certain  kind 


The  Notion  of  Number  33 

of  similarity.  Before  this  similarity  can  be  denied,  the  objects 
must  be  brought  together  for  comparison,  collected,  and  thus  a 
more  fundamental  kind  of  similarity  asserted.  Thus  the  plur- 
ality is  perceived  before  the  difference.  We  may  say  with 
Husserl,  that  differentiation  is  a  process  which  follows  upon 
analysis.  We  may  know  A  and  we  may  know  B,  but  it  is  not 
until  later  that  we  know  that  A  is  different  from  B.  Perception 
of  differences  is  not  the  same  as  different  perceptions. 

We  have  now  examined  enough  of  these  psychological  the- 
ories to  make  it  fully  apparent  that  a  confusion  of  standpoints 
pervades  the  great  majority  of  them.  Logic  is  so  entangled 
with  psychology  and  psychology  with  logic  that  the  results  are 
untenable  in  either  domain.  The  question  naturally  arises, — 
can  we  do  any  better?  Before  we  go  on,  it  might  be  well  to 
re-iterate  what  we  are  trying  to  do.  We  want  an  account  of 
the  type  of  act  which  by  repetition  leads  to  the  concrete  numbers 
of  practical  life  and  thence  to  the  abstract  concept  of  mathe- 
matics. 

§5.     Number  in  Terms  of  Behavior 

The  clue  to  our  number-behavior  must  be  found  in  the  fact 
that  man  is  subject,  not  only  to  mechanical  but  also  to  teleologi- 
cal  classification.  That  a  mere  collection  of  electrons  whose 
behavior  can  be  exactly  calculated  by  a  set  of  equations  could 
draw  arithmetical  conclusions  is  a  theory  which  may  surely 
be  disregarded.  It  is  with  the  possession  of  mind  that  we  rise 
above  the  mechanical  level. 

It  is  one  of  the  chief  characteristics  of  man  that,  as  we  have 
indicated  before,  he  is  able  to  behave  selectively,  that  is,  a  certain 
part  of  the  environment  is  singled  out  and  attended  to.  This 
process  is  essentially  teleological  and  governed  by  the  prevailing 
interests  of  the  person  in  question.  A  striking  example  of  this 
is  found  in  the  varying  comments  different  persons  make  upon 
the  same  presented  scene.  It  is  a  familiar  fact,  that  no  matter 
how  similar  physical  conditions  may  be,  one  person  will  observe 
objects  which  are  entirely  unnoticed  by  another.  This  ability 
to  concentrate  attention  is  essential  for  self-preservation;  and 
we  will  call  it,  the  "/Hs"  type  of  behavior.  Now  in  cases  of 
erroneous  perception,  hallucination  and  the  like,  the  selected 
"this"  cannot  be  said  to  exist,  for  our  very  discovery  that  it  is 
erroneous  is  due  to  the  attribution  of  contradictory  predicates 


34  The  Notion  of  Number  and  the  Number  of  Class 

to  the  object.  Consequently  existence  is  not  a  necessary  con- 
sequence of  this-ness.  That  it  is  behavior  toward  something, 
however,  is  undeniable;  we  shall  say,  accordingly,  that  each 
this  has  being,  although  not  necessarily  existence. 

As  another  condition  for  the  success  of  self-preservation,  man 
must  possess  a  kind  of  preparation  for  the  future  which  envelops 
him  like  a  fringe.  It  is  the  extension  of  the  past  into  the  future, 
the  result  of  the  fundamental  tendency  of  each  act  to  become  typical 
and  habitual.  This  casing  about  us  of  the  accustomed  response 
to  the  well-known  environment  enables  us  to  make  needed 
adjustments  rapidly  and  easily,  conserving  our  strength  for 
situations  demanding  concentrated  attention.  Every  manifes- 
tation of  hope,  belief,  or  fear  is  an  illustration  of  this  forward- 
looking  characteristic  of  human  behavior.  We  are  wound  up, 
so  to  speak,  like  a  spring,  and  always  charged  with  potential 
action.  Now  this  tendency  to  assume  the  attitude  of  tension, 
of  expectancy,  is  a  distinct  mode  of  behavior,  quite  different  from 
what  we  called  the  "this"  type.  This  new  element  we  will  call 
the  and  behavior. 

This  relation  is  implicitly  present  in  every  act  and  has  some 
properties  which  need  explaining.  The  connection  between 
the  situation  A  in  which  the  "and"  element  is  present,  and  the 
situation  B  to  which  it  is  connected,  is,  of  course,  the  "and" 
relation  itself.  Now  if  B  is  extremely  novel  and  striking,  if  A 
and  B  are  non-homogeneous,  the  connection  breaks  down  and  a 
new  type  of  act  presents  itself.  The  old  accustomed  reactions 
fail  to  work;  an  attitude  of  doubt  and  hesitation  is  assumed, 
until  when  the  pressure  becomes  great  enough,  man  exerts  his 
inherited  power  of  carrying  over  one  of  the  old  types.  This 
doubtful  behavior  will  be  called  the  "stop"  type.  The  condition 
that  it  should  arise  is  that  A  and  B  should  be  non-homogeneous. 

Let  us  pause  for  a  moment  and  recall  the  qualities  with  which 
we  have  found  man  to  be  endowed.  We  have  seen  that  in 
ordinary  experience,  our  behavior  is  of  the  type  "this"  and  "that" 
and  "the  other"  and  so  on,  until  a  "this"  is  reached  which  is 
sharply  distinct  from  the  preceding  with  reference  to  the  modes 
of  action  already  possessed  by  the  subject.  Here  we  come  to 
the  "stop."  We  may  state  the  typical  a  priori  form  of  cognitive 
experience  thus:  this  and  this  and  this  .  .  .  stop,  this  and 
.     .     .     stop,  etc. 


The  Notion  of  Number  35 

Here  we  have  in  the  most  rudimentary  form  of  cognition  the 
basis  for  numerical  behavior.  Let  us  suppose  a  man  gazing  at 
a  group  of  trees  and  his  behavior  of  the  type  "this  tree  A  and 
this  tree  B  and  this  tree  C,  stop"  and  let  us  also  suppose  the  act 
repeated  until  the  type  of  act  "A  tree  and  A  tree  and  A  tree" 
replaces  the  concrete  individual  act.  This  type  of  act  when 
further  abstract  and  carried  into  other  fields  becomes  "An  X 
and  an  X  and  an  X,"  finally  emerging  as  the  recognition  of 
the  abstract  number  three.  At  first  we  have  the  concrete  con- 
tents filling  the  places  between  the  "and"s;  next  they  are  replaced 
by  a  typical  term  representative  of  the  similarity  or  homogeneity 
of  the  objects;  finally  the  particular  kind  of  similarity  is  ab- 
stracted and  we  are  left  with  a  bare  set  of  homogeneous  "this"-es 
or  units.  The  next  stage  is  the  explicit  number,  obviously 
abstract  and  conceptual  in  character. 

This  is  enough  to  generate  a  number  system  such  as  a  Bra- 
zilian tribe  might  possess,  but  does  not  suffice  to  give  us  our  own 
scale.  We  may  now  introduce  the  rhythmical  sense,  which  we 
found  so  prominent  in  the  behavior  of  children  and  primitive 
men.  Manifestly  in  experience  abstractly  considered  the  stop 
or  break  may  occur  at  different  places  in  successive  situations; 
man  as  the  possessor  of  rhythmical  behavior  is  able  to  interpolate 
the  discontinuities  at  uniform  places.  In  this  way  a  collection 
can  be  grasped  in  groups  of  two's,  five's,  etc.  In  our  ordinary 
decimal  scale  the  interpolated  stop  comes  at  every  tenth  element. 

Not  only  does  this  principle  of  rhythm  enable  us  to  put  in 
the  breaks  but  it  provides  for  the  extension  to  the  general  num- 
ber system.  This  result  is  accomplished  by  the  change  from 
the  stop  externally  imposed  as  when  the  man  surveying  a  herd 
of  cows  comes  quite  suddenly  upon  a  bear  to  the  arbitrary  stop 
of  the  bank  teller  who  stacks  his  coins.  In  the  first  case  the 
homogeneity  of  the  known  objects  breaks  down  completely; 
in  the  second  the  self-imposed  rhythm  gives  a  motive  power 
which  bridges,  so  to  speak,  the  gap — very  much  as  dancers 
keep  step  during  a  rest  in  the  music.  We  come  then  to  the 
possibility  of  an  experience  in  which  all  the  breaks  are  inter- 
polated and  any  non-homogeneity  can  be  neglected.  Put 
abstractly,  this  would  be  "this  arid  this  and  this,  &  this  and  this 
and  this,  &  this  and  this  and  this.  .  .  .  After  any  given 
trio,  the  rhythm  of  the  process  carries  us  on.     Here  we  have  only 


36  The  Notion  of  Number  and  the  Notion  of  Class 

two  main  principles  at  work,  "this"  with  "and"  and  the  proviso 
that  no  real  "stop"  interfere.  "This  is  not  the  same  as  if  we 
should  attempt  to  count  a  homogeneous  set  with  an  inadequate 
supply  of  numerals  and  should  be  compelled  to^ apply  the  term 
"heap"  or  "many"  to  it.  That  act  was  essentially  of  the  stop 
variety,  whereas  the  assumed  proviso  in  the  second  case  directly 
forbids  such  hazy  behavior  and  gives  the  clear  conception  of  a 
never-ceasing  array  of  isolated  objects,  which  are  necessarily 
homogeneous.  This  new  conception  involved  a  new  principle, 
the  abstraction  of  the  external  stop,  and  is  therefore  essentially 
different  from  the  experiences  which  gave  the  ordinary  numbers. 

Two  other  abstractions  are  possible.  The  question  now 
arises,  what  is  to  be  done  with  the  potentiality  of  abstracting 
the  "and"}  If  this  is  granted  to  be  conceivable  we  have  be- 
tween the  initial  "this"  and  the  final  "stop"  a  collection  of 
"this"-es  not  disconnected  by  "stop"s  and  not  connected  by 
"and"s.  Although  they  are  distinct  from  one  another,  with 
the  exception  of  the  first  and  last,  they  are  not  distinguishable 
one  from  another  and  are  not  separated.  Of  no  individual 
"this"  can  we  say  that  it  has  a  next  following  or  a  next  preceding; 
the  isolation  has  been  abstracted.  It  follows,  therefore,  that 
if  we  consider  the  two  elements  which  we  can  distinguish,  namely 
the  first  and  last,  some  connection  must  replace  the  "and"; 
otherwise  they  could  not  belong  to  the  same  part  of  experience. 
This  connection  can  only  be  the  collection  of  elements  between 
the  first  and  last.  If  by  the  principle  of  rhythm  we  interpolate 
a  "stop"  in  the  set  and  are  thereby  enabled  to  distinguish  a 
third  "this"  between  this  new  element  and  each  end  there  must 
be  a  similar  connecting  collection.  The  group  of  "this"-es  is 
consequently  compact,  i.  e.,  between  any  two  distinguishable 
elements  there  is  always  a  third.  It  is  moreover  cohesive;  there 
are  no  finite  gaps  which  might  be  filled  by  "and."  It  has,  how- 
ever, one  further  property  which  is  of  essential  importance. 
Let  us  consider  the  possibilities  of  interpolation.  They  are 
manifestly  of  the  form  "this"  and  "this"  and  "this"  .  .  with 
nothing  to  destroy  their  homogeneity,  consequently  can  be 
considered  a  collection  of  the  kind  where  the  "stop"  is  abstracted 
and  the  "and"  left  in.  What  then  is  the  difference  between  the 
whole  set  and  its  subset? 

The  distinction  seems  to  be  this — that  no  matter  how  far 
our  interpolations  be  carried,  there  is  always  a  remainder  which 


The  Notion  of  Number  37 

at  any  given  step  is  perfectly  determinate;  that  there  is  a  re- 
mainder totally  outside  all  interpolations,  indeterminate  and 
containing  no  distinguishable  element,  and  around  each  inter- 
polated element  on  either  side  cluster  the  elements  of  the  re- 
mainder. Any  subset  of  the  interpolated  elements  is  deter- 
minate; this  is  not  true  for  the  whole  collection,  for  the  remainder 
is  not  determinate.  We  shall  call  such  a  set  where  the  "and" 
is  abstracted,  a  continuous  segment.  If  the  "stop"  now  is  ab- 
stracted we  get  the  notion  of  a  continuous  manifold  or,  in  brief, 
a  continuum.     This,  however,  comes  later. 

If  we  let  the  "and"  in  and  abstract  the  "this"  we  get  nothing 
in  particular — vague  awareness  but  no  actual  behavior.  This 
represents  the  numerical  zero  which  is  far  from  being  absolutely 
nothing.  Bergson  devotes  himself  to  some  length  in  his  "Crea- 
tive Evolution"  in  showing  that  the  image  of  zero  is  a  psycho- 
logical impossibility.  From  our  point  of  view  this  is  quite 
needless  for  apprehension  of  Zero  does  not  represent  the  absence 
of  experience,  but  the  absence  of  purposive,  selective  behavior. 
Bergson  would  put  the  Zero  in  the  object;  our  view  places  it 
in  the  subject. 

So  far,  therefore,  we  have  gained: 

(1)  A  group  of  types  of  indefinite  extent,  corresponding  to  the 
different  possible  positions  of  the  "stop." 

(2)  The  type  with  the  "stop"  abstracted — definite  infinity. 

(3)  The  type  with  the  "and"  abstracted — continuous  seg- 
ment. 

(4)  The  type  with  the  "stop"-"and"  abstracted — continuum. 

(5)  The  type  with  the  "this"  abstracted — zero. 

So  far  as  can  be  seen,  the  possibilities  of  abstraction  are  ex- 
hausted. Now  (2)  (3)  (4)  and  (5)  represent  a  different  level  of 
behavior  from  the  different  types  of  (1)  and  it  is  possible  for  a 
man  to  possess  a  great  variety  of  such  modes  of  behavior  with- 
out having  gained,  let  us  say,  such  an  abstraction  as  zero  or 
infinity.  Infinity,  for  example,  only  comes  with  the  feeling  of 
the  inherent  rhythm  of  the  number  series. 

So  much  for  the  psychological  origin  of  number.  Now  where 
does  mathematics  come  into  the  process?  We  might  say  that 
a  group  of  objects  possess  a  certain  number  when  our  behavior 
towards  the  group  is  of  such  and  such  a  type.  This  is,  however, 
only  Husserl's  fitting  of  forms  over  again  and  one-one  corre- 
spondence a  little  disguised.     The  goal  of  separating  number  and 


38  The  Notion  of  Number  and  the  Notion  of  Class 

correspondence  is  illusory  and  offers  small  advantages  if  gained. 
Cantor,  for  example,  defined  the  number  of  an  aggregate  as  the 
concept  which  is  attained  by  abstraction  of  the  nature  of  the 
elements  and  the  order  in  which  they  are  given.  If  unity  is 
substituted  for  each  element,  this  new  aggregate  is  regarded  by 
Cantor  as  a  symbolical  representation  of  the  number.  This 
view  is  substantially  one  which  would  make  the  different  num- 
bers our  abstract  forms  "this"  and  "this"  and  "this,"  etc.  We 
however  regarded  the  actual  number  as  gained  on  the  next 
higher  level  of  abstraction;  having  the  number  being  in  fact 
that  qualify  in  the  group  which  renders  the  application  of  the 
form  possible. 

Such  a  theory  might  seem  natural  enough  and  an  easy  con- 
sequence of  the  psychology  involved — yet  one  question  has 
been  completely  begged.  The  very  fundamental  assumption 
made  in  our  discussion  of  the  strata  of  number-perception 
(from  one  "stop"  to  the  next)  was  that  the  objects  should  be 
homogeneous.  What,  now,  is  the  significance  of  the  term 
"homogeneous"?  If  we  say  a  new  object  is  homogeneous  with 
the  old  when  it  is  similar  to  them,  the  problem  is  merely  re- 
moved, not  solved.  If  we  define  the  homogeneous  as  that  which 
occurs  between  two  "stop"  acts  we  fall  into  a  vicious  circle. 
The  real  state  of  affairs  is  that  to  define  the  psychological  nature 
of  number  we  have  been  obliged  to  assume,  in  some  measure 
at  least,  a  grouping  of  the  objects  of  experience  independent 
of  and  prior  to  an  observing  mind.  Now  such  an  "independent 
grouping"  is  nothing  more  nor  less  than  the  familiar  fact  that 
the  objects  of  the  universe  are  arranged  in  classes.  The  term 
homogeneous  refers,  therefore,  to  membership  of  the  same  class. 

Here  again  as  with  the  philosophers  as  Bosanquet  and  Sig- 
wart  and  the  logisticians  Frege  and  Russell,  from  an  attempt 
to  explain  the  notion  of  number  we  are  brought  face  to  face 
with  the  notion  of  class.  Now  this  notion  is  not  to  be  accepted 
blindly  as  an  indefinable,  if  our  short  experience  of  Russell's 
bewildering  treatment  of  the  null-class  is  to  serve  as  a  sample. 
To  sum  up  our  discussion  thus  far,  we  can  define  number  neither 
philosophically,  logically,  nor  psychologically  without  presup- 
posing the  notion  of  class.  To  presuppose  that  notion  blindly 
is  the  extreme  of  rashness,  consequently  we  are  forced  from 
the  question  "What  are  numbers"  to  the  allied  problem,  "What 
are  classes?" 


PART  TWO 

C.    The  Notion  of  Class 
§i.     The  Viewpoint  of  the  Mathematician 

The  ordinary  theory  of  the  unreflective  mathematician  in 
regard  to  classes  is  that  of  pure  extension.  When  he  mentions 
a  point-set,  he  means  just  the  several  individual  points,  which 
seem  to  group  themselves  into  a  set  by  a  sort  of  pre-established 
harmony.  The  class  is  simply  its  elements;  if  it  has  only  one 
member,  it  is  identical  with  that  member  and  most  mathe- 
maticians would  view  a  class  with  no  members  as  fictitious, 
if  not  absolutely  absurd.  This  position  is  rather  naive  and  if 
adhered  to  entails  all  manner  of  complications.  For  example, 
on  such  a  theory,  how  can  any  statement  be  made  about  infinite 
classes?  The  very  essence  of  an  infinite  group  is  that  it  cannot 
be  given  in  its  extension.  Moreover,  the  mathematician  is 
constantly  using  the  determining  law  of  the  group,  although 
unconscious,  perhaps,  of  its  full  importance.  Couterat  at  one 
time  asserted  boldly  that  mathematical  logic  could  exist  only 
if  the  standpoint  of  extension  were  used;  but  this  would  allow 
a  class  to  be  defined  only  by  enumeration  of  its  terms  and  would 
fail  to  account  for  an  infinite  series  constituting  a  whole,  in 
that  the  n-th  term  is  given  by  a  definite  formula.  It  is  no 
doubt  true  that  the  part  of  mathematics  known  as  the  theory 
of  aggregates  ("  Mengenlehre")  regards  an  "aggregate," — which 
is  only  a  logical  class  under  another  name, — as  constituted  by 
the  actual  catalog  of  the  elements.  Here  we  have,  let  us  say, 
a  case  of  objects,  or  terms  connected  by  "and."  As  plural,  not 
considered  as  forming  a  single  whole,  we  shall  call  such  an  enum- 
eration of  term,  a  collection. 

As  we  have  indicated,  it  is  impossible  to  rest  with  collections 
if  the  entire  range  of  mathematics  is  to  be  traversed,  for  infinite 
classes  do  occur  most  frequently.  Moreover,  we  may  as  well 
state  once  and  for  all  that  any  such  view  as  that  of  Russell's 
in  the  following  passage:  "Particular  classes,  except  when  they 
happen  to  be  finite,  can  only  be  defined  intensionally,  i.  e.,  as 
the  objects  denoted  by  such  and  such  concepts.  I  believe  this 
distinction  to  be  purely  psychological ;  logically,  the  extensional 
definition  appears  to  be  equally  applicable  to  infinite  classes, 

39 


4-0  The  Notion  of  Number  and  the  Notion  of  Class 

but  practically,  if  we  were  to  attempt  it,  death  would  cut  short 
our  laudable  endeavor  before  it  had  attained  its  goal," — is  not 
that  of  the  present  discussion.  The  complete  enumeration  of 
an  infinite  collection  is  to  be  regarded  by  us  not  only  as  repug- 
nant to  psychology,  habit  and  commonsense  but  absolutely  and 
finally  as  a  flat  logical  contradiction,  to  be  rejected  by  both 
mathematics  and  philosophy.  Any  treatment  of  infinite  classes 
must  necessarily  involve  an  intensional  element.  What  is  more 
than  this,  infinite  classes  enter  into  mathematics  in  so  many 
ways,  that  the  point  of  view  of  extension  is  clearly  inadequate 
and  to  insist  upon  it  would  be  pure  affectation. 

§2.     Frege's  Theory  of  Ranges 

It  is  one  of  the  chief  merits  of  Frege  to  have  called  attention 
to  this  breakdown  of  the  theory  of  extension,  and  to  have  advo- 
cated most  earnestly  the  rival  doctrine.  He  was,  however,  not 
so  much  influenced  by  the  consideration  of  infinite  classes  as 
by  desire  to  define  One  and  Zero.  The  null-class  manifestly 
does  not  occur  in  extension,  and  it  is  obvious  that  if  the  null- 
class  is  absurd,  Zero  is  no  better. 

His  most  important  contribution  to  the  refutation  of  the 
extensional  theory,  however,  is  in  the  case  of  the  solitary  ele- 
ment, which,  viewed  from  the  extensional  point  of  view  is  identi- 
cal with  its  containing  class.  This  has  been  put  in  the  following 
form:  "  If  a  is  a  class  of  more  than  one  term,  and  if  a  is  identical 
with  the  class  whose  only  member  is  a,  then  to  belong  to  a,  is 
to  belong  to  a  class  whose  only  member  is  a  and  hence  is  to  be 
identical  with  a.  Hence  a  is  the  only  member  of  a,  which  con- 
tradicts the  hypothesis  that  a  is  a  class  of  more  than  one  term." 
This  discussion,  if  correct,  absolutely  disposes  of  the  theory  of 
extension,  in  its  general  aspect. 

Frege's  positive  theory  of  classes  is  one  of  great  complexity 
and  difficulty.  It  depends  mainly  upon  the  primitive  ideas 
function  and  truth-value.  A  function  is,  roughly  speaking,  the 
invariant  part  in  some  expression,  where  we  regard  some  sign 
or  set  of  signs  as  being  replaceable  by  something  else.  In 
x2-|-x  we  do  not  have  a  function;  the  function  is:  (  )2-f(  ) 
and  the  x  is  the  argument.  When  we  have  a  logical  function 
(Russell's  propositional  function),  it  can  only  have  the  values 
truth  or  falsehood  and  is  the  same  as  a  BegrifT  or  concept.      If 


The  Notion  of  Class  41 

a  satisfies  the  function — if  (J) (a)  is  true — then  a  falls  under  the 
concept.  Objects  (Gegenstande)  are  contrasted  with  functions 
— anything  which  is  not  a  function  is  an  object.  Examples 
of  objects  are  numbers  and  truth-values.  The  name  of  an 
object  is  a  proper  name. 

We  are  now  ready  for  the  introduction  of  ranges  (Wertver- 
laiife).  Two  functions  are  said  to  have  the  same  range  when 
they  always  have  the  same  value  for  the  same  argument.  In 
the  case  of  truth-or-falsehood4unctions  we  can  replace  function 
by  concept  and  range  by  our  old  friend  "Umfang"  for  example, 
"the  concept  'square  root  of  four'  has  the  same  Umfang  as  the 
concept  'that  whose  square  multiplied  by  three  is  twelve' "  means 
the  same  thing  as  "'x2  =  4'  has  the  same  range  as  '3X2=I2.'" 

Suppose  we  have  now  the  proposition  "For  all  values  of  x, 
(J)(x)=4^(x)."  The  notion  of  ranges  enables  us  to  change  the 
universality  of  the  equation  into  a  range-equality.  Thus 
x(<J)(x))=x(<{/(x))  in  Frege's  notation.  How  important  this 
step  is  may  be  judged  from  his  own  language.  "This  substitu- 
tion may  be  regarded  as  unimportant.  .  .  .  To  such  I  may 
recall  that  in  my  "Grundlage  der  Arithmetik"  I  have  defined 
number  as  the  Umfang  of  a  concept."  "The  introduction  of 
the  symbol  for  ranges  seems  to  me  one  of  the  most  notable  exten- 
sions of  my  symbolism." 

The  notion  of  range,  however,  still  leaves  something  to  be 
wished.  The  complete  "Grundgesetze"  gives  us  some  infor- 
mation which  it  might  be  well  to  summarize: 

(1)  Numbers  are  ranges  and  are  applicable  to  ranges. 

(2)  Ranges  are  chiefly  useful  in  reducing  the  order  of  a  func- 
tion. 

(3)  Falsehood  is  the  range  of  |(<ce=e) 

(4)  Truth  is  the  range  of  — x. 

(5)  "The"  a  where  a  is  a  concept  with  more  than  one  par- 
ticular under  it  indicates  the  range  of  a.  "The  horse  is  an  intelli- 
gent animal"  gives  information  about  the  range  of  the  concept 
"horse."  "The  present  King  of  France"  denotes  the  range  of 
e(t£=e)  or  falsehood. 

(6)  If  a  range  has  only  one  member,  it  is  to  be  distinguished 
from  that  member. 

We  may  now  pass  on  to  the  "Nachwort"  of  the  "Grundge- 
setze" (pub.  1903)  where  ranges  come  again  to  the  foreground 


42  The  Notion  of  Number  and  the  Notion  of  Class 

in  a  most  perverse  and  disturbing  fashion.  It  seems  that  Russell 
communicated  to  Frege  (while  the  second  volume  of  the  "  Grund- 
gesetze"  was  on  the  press)  the  results  of  his  well-known  contra- 
diction that  the  class  of  classes  not  members  of  themselves  both 
is  and  is  not  a  member  of  itself.  The  "  Xachwort"  is  the  result 
of  this  communication.  It  opens  with  the  rather  dramatic 
words:  "Einem  wissenschaftlichen  Schriftsteller  kann  kaum 
etwas  Unerwunsch teres  begegnen,  als  dass  ihm  nach  Vollendung 
einer  Arbeit  eine  der  Grundlage  seines  Baues  erschuttert  wird." 

Frege  proceeds  to  give  an  account  of  Russell's  discovery, 
saying  that  something  belongs  to  a  class  ("Klasse")  if  it  falls 
under  the  concept  whose  Umfang  is  the  class.  The  paradox 
is  readily  elicited  and  various  ways  out  of  the  difficulty  sug- 
gested. Since  all  these  mean  substantial  modifications  in  the 
original  notion  of  range  we  must  consider  them: 

(i)  It  is  possible  to  hold  that  the  law  of  Excluded  Middle 
does  not  hold  for  classes.  This  would  deny  that  a  class  was  an 
object  while  about  it  there  is  nothing  unsatisfied  (argument 
place)  which  would  destroy  its  objectivity.  If  we  suppose 
classes  to  be  pseudo-objects  ("uneigentliche"),  it  must  be  speci- 
fied just  what  functions  can  take  these  pseudo-objects  for  argu- 
ments; for  there  certainly  is  at  least  one  property  which  classes 
have,  viz.,  Identity.  An  interminable  vista  of  distinctions 
opens  up,  and  we  may  well  despair  of  giving  a  complete  law, 
for  determining  what  objects  are  permissible  for  any  arbitrary 
function. 

(2)  Another  method  would  be  to  hold  that  classes  are  only 
fictions  (Scheineigennamen)  which  do  not  have  an  indication. 
They  would  then  be  parts  of  symbols,  which  would  only  indicate 
something  in  connection  with  the  whole — incomplete  symbols. 
Such  a  notion  is  nothing  surprising  or  new.  The  very  sim- 
plicity of  every  undefined  symbol  requires  that  the  parts  to  be 
detected  in  it  do  not  mean  anything  by  themselves.  However, 
what  does  the  theory  that  classes  are  such  parts  lead  to?  We 
would  have  to  admit  that  what  we  are  accustomed  to  regard  as 
numerals  would  not  be  symbols,  but  only  meaningless  parts 
of  symbols.  An  explanation  of  the  symbol  "2"  would  be  impos- 
sible; instead  of  this  we  would  have  to  explain  many  symbols 
which  would  contain  "2"  as  a  part,  but  are  not  logically  the 
sum  of  2  and  another  part.     It  would  then  be  illegitimate  to 


The  Notion  of  Class  4^ 

represent  such  a  part  by  a  single  letter,  since  by  itself  it  is 
meaningless.  With  this  the  universality  of  the  laws  of  arith- 
metic disappears,  along  with  such  expressions  as  ''number  of 
classes,"  "number  of  numbers." 

(3)  Since  these  paths  seem  to  lead  nowhere,  Frege  is  forced 
to  retrace  his  steps  and  investigate  the  matter  on  the  basis  of 
his  own  postulates.  He  gives  a  symbolic  statement  of  the  paradox 
entirely  without  the  use  of  the  epsilon  relation  and  uses  postulates 
V(b)  and  11(b),  1(g),  111(a).  It  is  in  one  of  these  that  the 
inconsistency  must  lurk  therefore,  1(g)  is  to  the  effect  that  a 
and  not-a  imply  not-a.  11(b)  is  to  the  effect  that  what  is  true 
of  all  first  order  functions  with  one  argument  is  true  of  any 
particular  one  of  them.  111(a)  says  that  if  a  =  b,  and  b  has 
any  property  f,  a  also  has  this  property. 

If  any  of  these  foregoing  postulates  be  denied,  it  is  hard  to 
see  how  any  formal  reasoning  whatever  could  survive;  how 
Contradiction,  Identity,  or  the  Syllogism  could  be  ever  legiti- 
mately employed  as  logical  terms.  The  trouble  therefore  must 
lie  in  Yb.  Postulate  V,  we  recall,  assumed  the  equivalence  of 
equality  of  ranges  with  universal  equality  of  functions  and  was 
the  basis  for  the  argument  that  even*  range  has  an  indication. 
However  Yb,  which  is  suspected,  is  not  so  strong;  it  is  merely 
half  of  the  equivalence,  i.e.,  that  the  equality  of  ranges  implies 
the  equality  of  functions.  With  its  downfall,  comes  the  possi- 
bility that  there  are  concepts  which  possess  no  Umfang  and 
that  the  expression  iij>(e)  has  no  indication — "und  doch  ist 
eine  solche  fur  die  Begrtindung  der  Arithmetik  unentbehrlich." 
Without  Y(b)  ranges  seem  to  have  vanished  from  the  substantial 
world  and  have  become  trifles  light  as  air. 

With  praiseworthy  courage  Frege  seeks  to  repair  the  damage. 
He  attempts  to  generalize  the  argument,  so  that  instead  of  going 
out  from  Yb  and  arriving  at  a  contradiction,  the  falsehood  of 
\~b  will  be  the  end  attained. 

This  effectually  disposes  of  (Y)b  at  one  fell  swoop,  and  proves 
moreover  from  the  very  generality  of  the  argument  that  it  is 
impossible  to  give  "range"  or  "Umfang"  such  a  meaning  that 
that  half  of  postulate  V  could  be  accepted.  What  is  then  to  be 
done?  By  using  a  different  method  of  approach,  Frege  is  able 
to  show  that  there  are  two  concepts  which  give  the  same  value 
when  used  as  arguments  for  the  general  second-order  function 


44  The  Notion  of  Number  and  the  Notion  of  Class 

M  ((}>),  and  under  one  of  these  this  value  falls  and  under  the 
other  it  does  not.  The  exception  which  makes  all  the  trouble 
is  clearly  the  Umfang  itself. 

Since  all  this  analysis  was  made  on  the  basis  of  unquestioned 
postulates,  the  conclusion  may  be  accepted.  It  suggests  clearly 
that  the  criterion  for  equality  of  ranges  must  be  revised  and 
indicates  a  method  for  evading  the  contradiction.  Frege  states 
the  revised  criterion  as  follows:  "The  Umfang  of  a  concept  f 
is  equal  to  that  of  a  concept  g  if  every  object  with  the  exception 
of  the  Umfang  of  f,  which  falls  under  f,  falls  under  g  and  also 
if  every  object  except  the  Umfang  of  g  which  falls  under  g  also 
falls  under  f."  This  leaves  V(a)  unaltered,  but  changes  the 
suspected  Vb  by  the  addition  of  two  exceptions.  It  is  now 
easy  to  show  that  the  contradiction  is  avoided. 

We  have  now  plunged  through  Frege's  doctrine  of  classes 
and  if  all  has  not  been  transparently  clear,  we  have  company  in 
our  misery.  Bertrand  Russell  himself  confesses  quite  frankly: 
"The  chief  difficulty  which  arises  in  the  above  theory  of  classes 
is  as  to  the  kind  of  entity  that  a  range  is  to  be.  .  .  .  It  would 
certainly  be  a  very  great  simplification  to  admit,  as  Frege  does, 
a  range  which  is  something  other  than  the  whole  composed  of 
the  terms  satisfying  the  propositional  function  in  question ;  but 
for  my  part  inspection  reveals  to  me  no  such  entity.  .  . 
Frege's  notion  of  a  range  may  be  identified  with  the  collection 
as  one,  and  all  will  then  go  well.  But  it  is  very  hard  to  see  any 
entity  such  as  Frege's  range,  and  the  argument  that  there  must 
be  such  an  entity  gives  us  little  help.  ...  It  would  seem 
necessary,  therefore,  to  accept  ranges  by  an  act  of  faith,  without 
waiting  to  see  if  there  are  such  things." 

The  status  of  ranges  seems  to  be  this — we  have  here  a  symbol 
which  may  be  significantly  said  to  have  properties  (in  Frege's 
usage,  possess  an  indication).  It  is  further  determined  by 
certain  postulates.  The  justification  of  this  indefinable  seems 
to  rest  upon  an  interpretation  of  the  word  range  (as  limited  by 
postulates)  in  some  sense  consistent  with  experience,  which  may 
be  limited  to  some  previous  objective  doctrine  of  logic,  mathe- 
matics, or  philosophy.  Frege  himself  offers  no  such  interpre- 
tation, which  fact  proves  one  of  two  conclusions:  either  (a)  he 
could  hit  upon  no  justification  of  the  symbol  or  (b)  the  notion 
had  become  so  familiar  to  him  through  long  use  that  he  could 
not  imagine  failure  to  grasp  it. 


The  Notion  of  Class  45 

Even  if  ranges  were  interpreted,  which  they  are  not,  other 
difficulties  arise  in  connection  with  Frege's  logical  system.  The 
first  of  these  was  noticed  by  Kerry  and  arises  in  connection  with 
the  theory  of  Concepts  (Begriffe),  Objects  (Gegenstande),  and 
Proper  Names.  Frege's  distinctions  are  so  delicate  in  regard 
to  these  matters  that  he  is  extremely  liable  to  misconstruction. 

Frege  holds  that  a  concept  cannot  be  made  a  logical  subject 
while  an  object  can  be  so  used;  in  his  own  words,  "Begriff  ist 
ein  mogliches  Pradicat  eines  singularen  beurteilbaren  Inhalts: 
Gegenstand  ein  mogliches  Subject  eines  solchen. "  The  names 
of  objects  are  used  with  the  definite,  of  concepts  with  the  indef- 
inite article.  Kerry  thinks  that  this  position  is  untenable.  If 
we  mean  by  subject  grammatical  subject  such  an  example  as: 
"The  concept  of  which  I  was  just  now  speaking,  etc.,"  is  suf- 
ficient to  refute  it  Here  we  have  a  concept  used  as  the  subject. 
He  goes  on  to  argue  that  concepts  can  be  shown  to  be  objects 
in  general,  for  we  find  no  difficulty  in  speaking  of  a  concept  of 
concepts  or  of  the  subordination  of  one  concept  to  another. 
Take  for  instance,  "The  concept  horse  is  an  easily  attained 
concept."  Here  we  have  both  subject  and  predicate  concepts 
with  no  object  entering  at  all. 

Kerry's  objections  were  answered  by  Frege  in  an  article 
"Uber  Begriff  und  Gegenstand."  He  contends  that  even  if 
Kerry's  criticisms  were  themselves  correct,  the  possibility  of 
the  existence  of  a  Begriff  which  could  not  be  a  subject,  and  a 
Gegenstand  which  could  not  be  an  object,  must  be  recognized, 
the  distinction  being  available  as  needed.  What  is  of  more 
weight,  however,  is  his  claim  that  Kerry's  argument  is  mistaken. 
While  it  is  of  course  true  that  we  may  have  a  concept  falling 
under  another  concept,  this  must  not  be  taken  to  mean  that 
the  concept  itself  is  used  as  the  subject.  Here  we  have  the  proper 
name  of  the  concept  not  the  function  itself.  In  "the  concept 
horse  is  an  easily  gained  concept"  we  are  talking  about  the  name 
of  the  concept,  it  is  possible  to  indicate  this  usage  by  appropriate 
quotation  marks. 

To  make  plain  what  Frege  means  by  this,  we  shall  have  to 
revert  to  his  theory  of  proper  names,  where  he  and  Russell  are 
in  flat  disagreement.  He  distinguishes  between  the  meaning 
(Sinn)  and  the  Bedeutung  or  Indication  of  a  term;  this  distinc- 
tion is  interpreted  by  Russell  as  being  roughly  equivalent  to 


46  The  Notion  of  Number  and  the  Notion  of  Class 

the  difference  between  a  concept  as  such  and  its  denotation. 
Thus  "the  first  President  of  the  United  States"  and  "the  father 
of  his  country"  have  the  same  indication  but  not  the  same  mean- 
ing. If  we  wish  to  speak  of  the  meaning  of  a  term  we  must 
signify  this  by  notation,  otherwise  the  indication  is  considered 
to  be  meant.  "A  proper  name  expresses  its  meaning  and  indi- 
cates its  indication."  In  Frege's  theory,  every  concept  has  a 
proper  name  which  indicates  the  denotation  of  the  concept 
but,  as  name,  expresses  the  meaning  of  the  concept.  In  the 
subsumption  of  one  concept  under  another,  it  is  not  the  indi- 
cation of  the  former  but  its  meaning  that  we  are  concerned  with, 
hence  it  is  the  proper  name  itself  which  has  the  relation.  In  a 
footnote  we  find  the  following  significant  observation  ".  .  . 
und  nicht  'der  Begriff  $(£).'  Die  letzten  Worte  bezeichnen 
also  eigentlich  nicht  einen  Begriff  in  unserem  Sinne,  obwohl 
es  nach  der  sprachlichen  Form  so  aussieht."  Every  proper 
name  has  the  two  sides,  can  be  used  predicatively  as  well  as 
in  the  role  of  subject.  Truth  and  Falsehood  which  are  assumed 
to  be  objects  do  not  have  proper  names  as  such,  so  we  arbitrarily 
give  them  the  proper  names  ( — x)  and  (x  =  term  not  identical 
with  itself).  This  is  in  direct  conflict  with  Russell's  theory 
of  denotation,  as  we  shall  see  later. 

We  can  see  now  why  Frege  does  not  use  a  function  quad- 
ratically.  If  we  have  cj>[4>(x)],  the  argument  is  not  the  indi- 
cation of  the  function  but  its  proper  name;  consequently  the 
revision  of  postulate  V  appears  to  be  justified,  as  a  translation 
into  symbolism  of  the  distinction  between  meaning  and  indi- 
cation. The  significance  and  sweeping  importance  of  this 
distinction  seems  to  have  passed  over  Russell's  head. 

Russell  goes  on  to  give  what  he  thinks  a  destructive  example 
of  Frege's  notion  of  function,  as  independent  of  an  expressed 
variable.  Consider  the  identical  function  of  x,  namely  x,  if 
this  were  indicated  by  no  variable,  when  we  abstract,  there 
would  be  nothing  left,  hence  no  function.  Frege,  however, 
did  not  admit  x  as  a  function  of  x;  its  place  is  filled  by  ( — x) 
(x  assumed)  which  is  always  a  function  whether  x  is  itself  or 
not. 

We  have  now  gone  far  enough  to  appreciate  the  profundity 
and  breadth  of  Frege's  analysis.  His  deficiencies  are  largely 
due  to  failure  to  see  the  wide-reaching  implications  of  his  own 


The  Notion  of  Class  47 

distinctions.  He  has  re-introduced  meaning  into  logic  and 
sees  clearly  the  double  side  of  concepts,  and  proper  names, 
although  the  treatment  given  is  rather  formless.  Ranges  remain 
mysterious,  but  necessary  and  their  interpretation  a  problem 
for  our  own  undertaking.  All  in  all,  Frege  seems  to  have  walked 
unscathed  among  the  pitfalls  into  which  Russell  has  fallen  and 
to  have  maintained  an  unshaken  serenity  in  the  face  of  tre- 
mendous complications.  His  solution  of  the  paradox,  moreover, 
at  least  on  the  face  of  it,  is  consistent  with  his  general  logical 
doctrine.  The  neglect  of  his  contemporaries  may  have  been 
due  to  his  cumbrous  symbolism  and  the  obscurities  of  his  deli- 
cate analysis  but  we  may  look  to  the  future  to  avenge  him,  for 
it  would  be  scarcely  too  much  to  say  that  no  more  brilliant, 
subtle,  or  original  work  on  the  foundations  of  logic  has  appeared 
since  the  days  of  Aristotle. 

§3.     Russell's  Infinite  Variety 

When  we  turn  from  Frege  to  Bertrand  Russell,  it  is  very 
much  as  if  we  should  glance  away  from  the  regular  symmetry 
of  some  Greek  temple  and  contemplate  the  myriad  ramifications 
of  a  Gothic  cathedral.  The  thought  of  Frege  is  not  without  a 
certain  majestic  sweep  as  it  continually  rises  above  its  former 
incompleteness  and  develops  by  virtue  of  its  own  imperfections. 
With  Russell,  on  the  other  hand,  a  bewildering  variety  of  the- 
ories follow  upon  one  another  in  such  dazzling  profusion  that 
the  result  is  not  so  much  to  convince  the  reader  as  to  hypnotize 
him.  No  sooner  does  a  conclusion  receive  its  full  and  persuasive 
enunciation  than  the  possible  alternatives  and  objections  crop 
up;  to  be  repudiated  or  sympathized  with,  as  the  case  may  be. 
The  greater  part  of  the  "Principles  of  Mathematics,"  therefore, 
does  not  resemble  a  constructive  theory  so  much  as  a  Parlia- 
mentary debate,  and  gives  the  impression  that  its  author  is  a 
brilliant,  but  unsafe  guide  to  mathematical  philosophy. 

We  caught  some  hint  of  this  erratic  tendency  in  Russell  while 
we  were  considering  his  theory  of  the  null-class.  His  general 
theory  of  classes  suffers  the  same  changes  as  the  null-class. 
We  start  out  in  the  "Principles"  with  the  dictum  that  the  exten- 
sional  view  of  classes  is  necessary  for  symbolism  and  that  the 
failure  of  this  view  in  regard  to  infinite  classes  is  "purely  psy- 
chological."    However,  there  is  a  distinction  between  the  class 


48  The  Notion  of  Number  and  the  Notion  of  Class 

as  many  and  the  class  as  one,  and  this  distinction  may  be  enough 
to  solve  the  paradox,  although  it  does  not  distinguish  a  unit 
class  from  its  only  member.  The  notion  of  propositional  func- 
tion is  declared  to  be  the  genesis  of  classes.  Since  it  is,  as  a 
function,  neither  true  nor  false,  but  indeterminate  and  may  be 
true  for  certain  values  and  false  for  others.  Thus  a  class  is 
determined — a  collection  of  values  for  which  it  is  true.  Equiva- 
lent propositional  functions,  i.  e.,  true  for  the  same  value,  deter- 
mine the  same  class.  Russell  calls  this  point  of  view  modified 
extension.  The  many  variations  of  this  theme  in  the  "Prin- 
ciples" we  have  already  found  in  the  treatment  of  the  null  class. 

Under  the  pressure  of  his  paradox  and  Frege's  seemingly 
irrefutable  argument  that  the  unit  class  is  not  to  be  identified 
with  its  member,  Russell  is  compelled  to  change  his  theory, 
of  classes  in  the  same  volume,  while  considering  Frege's  ranges. 
He  says:  "Nevertheless  the  non-identification  of  the  class  with 
the  class  as  one  .  .  appears  unavoidable,  and  by  a  process  of 
exclusion  the  class  as  many  is  left  as  the  only  object  which  can 
play  the  part  of  a  class.  By  a  modification  of  the  logic  hitherto 
advocated  in  the  present  work,  we  shall,  I  think,  be  able  at 
once  to  satisfy  the  requirements  of  the  Contradiction  and  to 
keep  in  harmony  with  common  sense."  Back  in  the  sixth 
chapter  occurs  the  following:  "A  plurality  of  terms  is  not  the 
logical  subject  when  a  number  is  asserted  of  it;  such  propositions 
have  not  one  subject,  but  many  subjects."  This  seems  to  be 
directly  contradicted  by  the  identification  of  class  with  class 
as  many. 

Russell  goes  on  to  recapitulate  the  possible  theories  of  classes 
which  present  themselves.  His  enumeration  is  sufficiently 
important  for  our  purposes  to  justify  repetition.  A  class,  he 
says,  may  be  identified  with:  (i)  the  predicate;  (2)  the  class 
concept;  (3)  the  concept  of  the  class;  (4)  Frege's  range;  (5) 
the  collection  of  terms;  (6)  the  whole  composed  of  the  terms 
of  the  class.  The  first  three  are  inadmissible,  since  they  do  not 
permit  a  class  to  be  defined  by  the  enumeration  of  its  members 
Frege's  range  is  declared  to  be  incomprehensible.  A  collection 
cannot  be  a  class  because  it  is  grammatically  plural,  and  (6) 
is  refuted  by  the  argument  that  the  singular  class  is  not  to  be 
identified  with  its  only  member. 

Having  thus  demolished  all  previous  theories,  Russell  finds 
himself  in  a  quandary.     After  considering  and  rejecting  various 


The  Notion  of  Class  49 

alternatives,  he  declares  himself  thus:  "The  logical  doctrine 
which  is  thus  forced  on  us  is  this:  the  subject  of  a  proposition 
may  be  not  a  single  term  but  essentially  many  terms  .  .  but 
the  predicates  or  class  concepts  or  relations  which  can  occur 
in  propositions  having  plural  subjects  are  different  from  those 
having  single  terms  as  subjects.  Although  a  class  is  many  and 
not  one  .  .  classes  can  be  counted  as  though  each  were  a 
genuine  unity;  and  in  this  sense  we  can  speak  of  one  class.  It 
will  now  be  necessary  to  distinguish  (1)  terms;  (2)  classes;  (3) 
classes  of  classes  and  soon.  We  shall  hold  .  .  that  no  mem- 
ber of  one  set  is  a  member  of  another  and  that  x  s  u  requires 
that  x  should  be  of  a  set  a  degree  lower  than  the  set  to  which 
u  belongs." 

He  asserts  that  it  will  be  necessary  to  indicate  whether  the 
field  of  every  variable  is  terms,  classes,  or  classes  of  classes,  etc., 
arriving  finally  at  a  more  extensional  view  than  before,  finding, 
"that  the  class  as  many  is  the  only  object  defined  by  a  preposi- 
tional function  .  .  that  the  class  as  one  is  probably  a  genuine 
entity  except  when  the  class  is  defined  by  a  quadratic  function." 

Later,  in  Appendix  B,  a  slightly  different  point  of  view  is  set 
forth,  which  elaborates  the  theory  of  logical  types  previously 
hinted  at.  This  is  substantially  a  rigid  distinction  between 
terms,  classes  of  terms,  etc.  The  different  possibilities  are  rather 
carefully  examined  and  all  goes  well  until  §500  is  reached,  when 
the  dread  bugbear,  the  Contradiction,  crops  out  again,  this 
time  in  considering  propositions  as  a  type.  Thus,  although  the 
contradiction  of  classes  is  solved  by  the  doctrine  of  types,  a 
closely  analagous  difficulty  is  raised  in  the  doctrine  itself.  "It 
seems  possible,  of  course,  to  hold  that  propositions  themselves 
are  of  various  types  and  that  logical  products  must  have  propo- 
sitions of  only  one  type  as  factors."  This  alternative  is  de- 
nounced as  "harsh  and  artificial,"  consequently  the  notion  of 
class  remains  paradoxical  at  the  end  of  the  very  last  Appendix 
of  the  "Principles." 

The  next  important  sign  of  Russell's  increasing  passion  for 
extension  and  growing  distrust  of  the  notion  of  class  is  found 
in  his  article,  "On  Some  Difficulties  in  the  Theory  of  Transfinite 
Numbers  and  Order-Types."  Here  he  again  contemplates  the 
paradoxes  and  is  able  to  develop  a  great  budget  of  similar  ones 
by  a  common  recipe. 


5<3  The  Notion  of  Number  and  the  Notion  of  Class 

Three  ways  are  open,  he  finds,  for  the  logician  who  wishes  to 
avoid  paradox  and  contradiction.  They  are,  following  his 
designation:  (i)  the  zigzag  theory;  (2)  the  theory  of  limitation 
of  size;  (3)  the  no-classes  theory.  The  zigzag  theory  presumes 
that  logical  functions  determine  classes  when  they  are  simple 
"and  only  fail  to  do  so  when  they  are  complicated  and  recondite." 
Its  great  disadvantage  is  that  the  postulates  determining  which 
functions  are  to  determine  classes  and  which  shall  not,  are 
necessarily  complicated,  difficult  and  without  much  plausibility. 
Indeed  the  great  guiding  principle  in  the  adopting  of  any  definite 
axiom  of  this  kind  is  that  it  avoids  a  well  known  paradox.  This, 
our  author  admits,  "is  a  very  insufficient  principle,  since  it 
leaves  us  always  exposed  to  the  risk  that  further  deductions 
will  elicit  contradictions." 

The  theory  of  limitation  of  size  is  based  on  the  idea  that  it  is 
"size  which  makes  classes  go  wrong."  This  theory  becomes 
specialized  into  the  assumption  "that  a  proper  class  must  always 
be  capable  of  being  arranged  in  a  well-ordered  series."  The 
great  drawback  is  that  we  are  led  to  a  sort  of  trial  and  error 
process.  The  No-classes  theory  remains  to  be  considered,  and 
although  it  is  not  definitely  adopted  at  this  time,  is  evidently 
viewed  with  greatest  favor.  Its  procedure  is  brief,  stringent, 
and  "drastic."  Classes  and  relations  are  to  be  straightaway 
banished  from  logic;  everything  is  to  be  expressed  in  terms  of 
proposition,  function  and  truth  or  falsehood.  This  process  is 
admitted  to  be  complicated  and  difficult,  not  to  say  somewhat 
startling  to  common  sense.  A  degree  of  safety  is  claimed  for 
it,  however;  although  at  the  time  of  writing  (before  Nov.  24, 
1905)  it  is  not  espoused.  Russell  is  still  disposed  to  cling  doubt- 
fully to  the  ordinary  logic. 

A  footnote  dated  February  5th,  1906,  reveals  the  decided 
break:  "From  further  investigation  I  now  feel  hardly  any  doubt 
that  the  no-classes  theory  affords  the  complete  solution  of  all  the 
difficulties  stated  in  the  first  part  of  the  paper."  The  decision 
has  now  been  made.  Classes  are  banished,  and  Russell  has 
from  that  celebrated  "lair"  "midway  between  extension  and 
intension"  come  out  unqualifiedly  for  extension. 

The  character  of  these  investigations  was  revealed  in  articles 
in  the  "Revue  de  Metaphysique  et  de  Morale"  and  the  "Ameri- 
can Journal  of  Mathematics."     In  these  the  theory  of  types  of 


The  Notion  of  Class  51 

Appendix  B  is  united  with  the  no-class  theory  and  a  formal 
disproof  of  the  contradictions  attempted.  This  treatment 
appears  in  very  nearly  unaltered  form  in  the  "Principia,"  and 
its  abolition  of  classes  needs  careful  consideration.  Russell's 
point  is  that  classes  are  incomplete  symbols.  He  neglects 
absolutely  to  reply  to  Frege's  objections  to  this  supposition. 

On  p.  69  of  the  "Principia"  we  find  an  incomplete  symbol 
defined  as  a  symbol  which  is  not  supposed  to  have  any  meaning 
in  isolation  but  is  only  defined  in  certain  contexts.  Such  incom- 
plete symbols  are  to  be  used  differently  from  proper  names. 
Here  we  may  object.  Proper  names  by  no  means  have  a  con- 
stant meaning  independent  of  context.  We  can  agree  with 
Russell  when  he  says  that  "Socrates"  is  a  proper  name  which 
constantly  denotes  an  individual,  only  if  we  suppose  "Socrates" 
to  denote  the  physical  man;  if  we  discuss  Greek  history  and 
mention  the  Socrates  of  the  "Memorabilia"  and  the  Socrates 
of  the  Platonic  dialogs  and  the  youthful  Socrates  and  so  on, 
here  "Socrates"  has  anything  but  a  constant  meaning. 

Russell  goes  on  to  exhibit  a  kind  of  proposition  in  which  the 
subject,  while  purporting  to  be  a  proper  name,  because  preceded 
by  the  definite  article  "the"  seems  to  be  masquerading  under 
false  colors.  The  cited  case  is:  "The  round  square  does  not 
exist"  and  we  are  told  that  we  can  not  regard  this  proposition 
as  a  denial  of  the  existence  of  the  object  "round  square."  "For 
if  there  were  such  an  object  it  would  exist;  we  cannot  first  assume 
that  there  is  a  certain  object  and  proceed  to  deny  that  there  is 
such  an  object."  Where  the  asserted  proposition — such  an 
object  is  implies  that  such  an  object  exists" — comes  from,  is 
left  to  the  reader's  imagination.  The  two  words  are  confused 
throughout  the  passage  in  question.  To  disentangle  ourselves 
from  the  meshes  of  this  equivocation  we  shall  have  to  make  use 
of  a  distinction,  which  Russell  himself  was  once  very  scrupulous 
in  making. 

On  p.  449  of  the  "  Principles"  we  find  this  illuminating  passage. 
"Being  is  that  which  belongs  to  every  conceivable  term, — to 
everything  that  can  possibly  occur  in  any  proposition,  true  or 
false,  and  to  such  propositions  themselves.  .  .  .  Being 
belongs  to  whatever  can  be  counted.  To  mention  something 
shows  that  it  has  being.  Existence,  on  the  contrary,  is  the 
prerogative  of  some  only  among  beings.     To  exist  is  to  have  a 


52  The  Notion  of  Number  and  the  Notion  of  Class 

specific  relation  to  existence."  We  see  from  this  that  the  class 
of  existents  is  a  sub-class  of  beings.  A  round  square  is  denied 
by  the  proposition  to  fall  in  this  sub-class.  The  proposition 
"The  round  square  is  not"  is  an  incomplete  symbol;  "The  round 
square  does  not  exist"  is  not  such. 

If  we  accept  the  distinction  between  being  and  existence  (and 
surely  no  good  reason  has  been  advanced  for  discarding  it) 
the  situation  presents  no  difficulty  and  there  is  no  need  for 
analyzing  out  the  grammatical  subject  as  our  author  proceeds 
to  do.  The  argument  goes  on  to  show  that  such  "the"  phrases 
are  always  incomplete  symbols,  but  a  like  confusion  infests 
the  proof.  Here  the  illustration  is  "Scott  is  the  author  of 
Waverley"  and  the  argument  is  as  follows.  "Scott"  is  a  genuine 
proper  name,  and  "the  author  of  Waverley"  if  a  complete  symbol 
denotes  "Sir  Walter  Scott"  also,  giving  the  proposition  "Scott 
is  Scott"  which  is  denounced  as  trivial.  Hence  it  is  concluded 
summarily  that  "the  author  of  Waverley"  must  be  an  incom- 
plete symbol. 

We  may  object  in  two  ways,  (a)  The  principle  of  Identity 
is  not  so  trivial  as  is  commonly  supposed.  Boyce  Gibson  says 
in  his  "Problem  of  Logic" — "To  state  a  proposition  we  must, 
to  put  the  matter  quite  generally,  specify  our  meaning.  If  we 
wish  to  make  a  definite  statement  of  fact  we  must  first  specify 
that  aspect  of  the  total  topic  which  we  wish  particularly  to 
speak  about;  this  will  give  us  the  subject  of  our  statement.  We 
have  then  to  specify  this  subject  by  predicating  something  about 
it  that  is  other  than  itself.  The  Principle  of  Identity  will  be  a 
Principle  of  Identity  in  relation  to  difference."  Thus  what  we 
are  doing  in  a  statement  of  identity  is  connecting  contexts, 
(b)  Moreover  in  "the  author  of  Waverley"  and  "Scott"  we 
do  not  have  a  proper  name  and  a  denoting  phrase  but  two  denot- 
ing phrases  and  our  proposition  merely  asserts  their  equivalence, 
that  is,  says  that  they  have  the  same  denotation.  Why  our 
customary  proper  names  should  be  given  an  exalted  rank  is 
hard  to  see;  they  do  not  individuate  in  the  true  philosophical 
sense  any  more  than  denoting  phrases. 

Most  of  the  difficulty  is  due  to  Russell's  doctrine  of  individuals, 
which  is  in  some  respect  necessitated  by  the  theory  of  types. 
We  can  only  make  a  distinction  between  types  as  n  and  n+i  if 
there  is  a  first  type;  this  must  be  the  type  of  individuals.  We 
shall  return  to  this  later. 


The  Notion  of  Class  53 

Another  supposed  puzzle  is  exhibited  in  order  to  justify  the 
peculiar  theory  of  denotation.  This  is  "The  present  King  of 
France  is  bald."  The  present  King  of  France  not  existing, 
the  statement  is  paradoxical.  However,  we  may  reduce  this 
to  the  problem,  whether  we  are  allowed  to  introduce  non-ex- 
istents  into  arbitrary  existent  classes,  and  this  current  logical 
usage  appears  to  allow.  What  Russell  desires  to  do  is  to  convert 
the  proposition  into  "The  King  of  France  exists  and  is  bald," 
claiming  to  find  his  justification  in  ordinary  usage.  Ordinary 
usage  is  not  a  safe  guide  in  connection  with  existence,  however, 
for  it  is  prone  to  identify  existence  with  the  physical  world 
order,  and  the  circle,  qua  perfect  circle,  would  not  exist  any 
more  than  the  present  King  of  France. 

We  may,  for  example,  interpret  the  phrase  as  meaning  "the 
rightful  King  of  France"  and  thus  have  an  existent  subject. 
This  shows  that  a  phrase  may  not  denote  an  existent  when  we 
consider  a  certain  limited  context  or  interpretation,  but  often 
the  sphere  may  be  widened  enough  for  the  proposition  to  be 
definitely  interpreted.  No  statement  of  existence  can  legiti- 
mately be  imported  into  the  proposition.  What  we  have  is  the 
statement  of  membership  in  a  class.  If  we  hold  that  an  existent 
class  can  only  have  existents  as  members,  this  would  settle  the 
question,  but  that  is  another  problem. 

Classes,  we. are  told,  are  incomplete  symbols;  they  themselves 
do  not  mean  anything  at  all.  "Thus  classes,  so  far  as  we  intro- 
duce them,  are  merely  symbolic  or  linguistic  conveniences,  not 
genuine  objects,  as  their  members  are,  if  they  are  individuals." 
Russell  admits  that  no  proof  is  forthcoming  that  classes  are,  as 
stated,  incomplete  symbols  although  "arguments  of  more  or  less 
cogency"  are  adduced,  all  of  which  are  equivalent  to  the  assump- 
tion that  the  same  object  cannot  be  both  one  and  many,  which, 
observes  our  author,  "seems  impossible."  If  it  is  impossible, 
a  formal  proof  should  be  available;  if  it  is  mere  disbelief  and 
want  of  faith,  such  matters  of  psychology  should  not  be  allowed 
to  intrude. 

Given  two  propositional  functions  which  have  the  same  truth 
value  for  the  same  argument,  i.  e.,  are  formally  equivalent,  it 
seems  obvious  that  they  have  something  in  common, — namely 
that  which  we  would  naturally  suppose  to  be  the  class, — but 
we  are  forbidden  to  rush  to  the  conclusion  that  this  is  the  case. 


54  The  Notion  of  Number  and  the  Notion  of  Class 

"We  do  not  assume  that  there  is  such  a  thing  as  an  extension; 
we  merely  define  the  whole  phrase  'having  the  same  extension." ' 
What  Russell  does  is  to  say  that  say  that  any  function  f  of  a 
class  is  equivalent  to  the  corresponding  function  of  a  predicative 
function  formally  equivalent  to  the  one  which  defined  the  class. 
In  this  way  classes  are  regarded  as  having  been  avoided.  How- 
ever, this  process  of  defining  any  function  of  x  without  defining 
x  appears  to  be  invalid.  Let  us  recall  that  in  criticism  of  Frege, 
Russell  used  the  so-called  identical  function  of  x,  namely  x 
itself;  this  function  overthrows  his  evasion  of  the  notion  of  class. 
For  regarding  the  class  as  a  function  of  the  class  we  see  it  defined 
as  the  predicative  function,  which  is,  in  fact,  gained  by  the  axiom 
of  reducibility.  Here  is  really  an  extremely  intensional  view 
of  classes,  for  which  there  appears  little  philosophical  defense. 

The  existence  of  Individuals  is  deemed  of  fundamental  impor- 
tance. That  the  axiom  which  insures  it,  is  convincing,  is  hardly 
to  be  conceded.  For  if  any  term  is  an  individual,  it  cannot  be 
a  function  or  class  and  in  ordinary  usage  no  such  "principium 
individuationis"  appears.  The  facts  of  the  case  seem  to  be 
that  the  term  individual  is  essentially  relevant  to  the  universe 
of  discourse.  As  we  have  seen,  Socrates  is  in  one  "sense"  or 
universe  an  individuating  term,  in  another,  a  universal.  As  for 
the  absolute  individual,  which  is  to  be  a  universal  in  no  universe 
of  discourse — this  is  a  concept  which  appears  to  involve  insuper- 
able difficulties. 

As  far  as  considering  the  theory  of  types  to  be  a  disguised 
theory  of  the  universe  of  discourse,  we  have  seen  that  this  theory 
necessitates  a  first  type, — individuals — and  individuals  only 
exist  in  reference  to  a  universe  of  discourse.  How  can  the 
theory  of  types  explain  something  which  it  presupposes?  This 
problem  has  been  dexterously  avoided. 

Russell  then  has,  in  effect,  while  ostentatiously  rejecting 
classes,  regarded  them  as  equivalent  to  certain  propositional 
functions.  Frege  has  regarded  them  as  ranges,  vague  and 
unexplained  notions.  We  can  be  satisfied  with  neither.  Rus- 
sell's ideas  of  types  and  individuals  infect  his  theory  of  classes, 
and  Frege's  notions  are  lacking  in  justification.  We  shall  turn, 
therefore,  to  current  mathematical  usage,  for  clues  by  which 
to  investigate  the  notion  of  class,  with  which  the  number  con- 
cept seems  to  be  inseparably  connected. 


The  Notion  of  Class  55 

D.    Mathematical  Usage 

We  must  now  examine  the  most  formal  and  rigorous  procedure 
of  the  mathematicians.  They  customarily  hold  a  mathematical 
science  to  be  dependent  upon  (1)  the  undefined  symbols,  (2) 
the  general  axioms,  (3)  the  existence  postulates.  The  axioms 
provide  schemes  of  classification  of  the  undefined  symbols,  and 
suggest  means  for  defining  unwieldy  combinations  in  terms  of 
these  undefined  symbols.  After  investigating  the  various  pos- 
sible classifications  which  interest  him,  these  being  tabulated 
under  the  name  of  theorems,  the  mathematician  becomes  con- 
cerned with  a  critical  survey  of  his  initial  suppositions.  It  is 
important  to  note  that  so  long  as  he  is  merely  establishing  theorems, 
the  mathematician  is  only  connecting  arbitrary  symbols  (the  inde- 
finables)  by  the  means  of  arbitrary  rules — the  postulates. 

From  this  new  and  later  point  of  view,  however,  he  wishes 
to  investigate  his  postulates  with  a  view  as  to  their  consistency, 
categoricity,  and  mutual  independence.  Consistency  is  the 
consideration  of  first  and  foremost  importance.  If  a  set  of 
postulates  is  inconsistent,  it  must  be  revised  or  discarded.  What 
is  the  test  of  consistency?     Two  methods  are  generally  accepted: 

(1)  A  set  of  postulates  is  said  to  be  inconsistent  if  contra- 
dictory statements  can  be  made  about  the  undefined  symbols, 
these  statements  being  correctly  deduced  from  the  whole  or  a 
subset  of  the  assumptions  in  question. 

(2)  Such  a  set  is  said  to  be  consistent  if  some  interpretation 
of  the  undefined  symbols  can  be  found  which  satisfies  the  postu- 
lates. 

It  is  important  to  remember  that  these  are  the  only  criteria 
of  consistency  which  are  accepted.  If  we  look  at  them  more 
closely  we  can  see  that  both  (1)  and  (2)  directly  depend  upon 
the  ingenuity  and  perseverance  of  the  mathematician  who  is 
working  upon  the  system.  For  example,  a  contradiction  may 
lurk  hidden  among  the  implications  of  some  obscure  theorem, 
whose  possibilities  no  one  has  ever  thought  of  investigating. 
From  this  point  of  view  the  mathematician  is  always  walking 
upon  the  brink  of  a  precipice,  for,  no  matter  how  many  theorems 
he  deduces,  he  cannot  tell  that  some  contradiction  will  not  await 
him  in  the  infinity  of  consequences.  Let  us  suppose,  then,  that 
as  far  as  he  has  gone,  no  inconsistency  has  been  come  upon,  and 
being  dissatisfied  with  the  indefinite  conclusion  that  the  postu- 


56  The  Notion  of  Number  and  the  Notion  of  Class 

lates  may  or  may  not  be  consistent,  he  grasps  the  other  possi- 
bility, and  undertakes  to  demonstrate  their  consistency. 

Now  it  devolves  upon  him  to  find  an  interpretation  for  his 
arbitrary  words  such  that  the  postulates  still  hold.  The  only 
way  of  doing  this  is  by  recourse  to  the  accepted  conclusions  of 
some  other  set  of  postulates.  The  problem  is  thus  shifted  from 
one  domain  to  another,  and  the  only  thing  the  mathematician 
is  justified  in  saying  is  that  his  set  is  just  as  consistent  as  the  set 
in  which  the  interpretations  occur.  This  process  can  go  on 
until  he  can  establish,  it  may  be,  that  his  set  is  just  as  consistent 
as  the  axioms  for  the  natural  number  system.  Back  of  this 
the  non-philosophic  mathematician  does  not  care  to  go,  nor 
is  it  evident  that  continuing  the  search  would  lead  to  any  profit- 
able results. 

Now  our  survey  of  this  attempt  of  the  mathematician  to 
interpret  his  "words"  makes  it  plain  that  he  is  reducing  his 
system  to  a  system  involving  defined  combinations  of  the  unde- 
fined symbols  of  some  other  branch  of  the  science.  He  is  assum- 
ing the  formal  equivalence  of  his  symbols  to  these  particular 
combinations;  that  is,  he  feels  himself  justified  in  substituting 
the  "interpretation"  for  the  undefined  symbol  wherever  it 
occurs  and  vice  versa.  Suppose  now  that  no  such  scheme  of 
equivalences  occurs  to  the  mathematician.  It  is  entirely  con- 
ceivable that,  for  some  length  of  time,  the  science  may  contain 
no  inconsistency  and  yet  receive  no  interpretation,  being,  so 
to  speak,  in  a  delicate  position.  Moreover,  even  when  inter- 
preted to  its  fullest  extent,  it  is  just  as  consistent,  neither  more 
nor  less,  as  the  system  of  positive  integers.  That  the  number 
system  contains  no  contradiction  is  the  fundamental  rock  of 
mathematical  faith. 

The  question  of  categoricity  is  closely  allied.  We  call  a  set 
of  postulates  categorical  if  there  is  only  one  interpretation  to 
be  given  to  the  undefined  symbols  or  if  any  two  such  systems 
of  interpretations  can  be  made  simply  isomorphic.  Now  the 
task  of  examining  all  possible  interpretations  and  rejecting 
some  as  not  satisfying  the  postulates,  and  others  as  being  essen- 
tially and  abstractly  the  same  as  the  system  of  interpretations 
we  wish  to  consider  categorical,  is  not  a  process  which  can  be 
completed  in  any  finite  length  of  time.  As  in  the  case  of  con- 
sistency, no  matter  how  far  and  how  favorably  the  investigation 


The  Notion  of  Class  57 

progresses,  some  residuum  of  doubt  must  always  remain.  Such 
being  the  situation,  the  mathematician  is  regarded  as  having 
demonstrated  the  categoricity  of  his  set  of  postulates  when  he 
has  proved  that  they  possess  the  categoricity  of  the  axioms  for 
the  number  system.  The  consideration  of  independence  forces 
us  to  a  somewhat  different  point  of  view.  Here  we  wish  to  show 
that  each  assumption  in  turn  cannot  be  deduced  from  all  the 
rest;  in  other  words,  that  no  one  of  the  axioms  is  redundant. 
This  is,  in  effect,  the  opposite  of  categoricity.  The  more  postu- 
lates used,  the  more  they  will  tend  to  be  categorical  and  not 
independent.  The  happy  medium  is  the  goal  of  the  mathe- 
matician— a  set  which  contains  no  redundancies,  and  is  abso- 
lutely categorical.  An  assumption  Pi,  let  us  say,  is  said  to  be 
independent  of  the  other  assumptions  Pa  .  .  Pn  when  an 
interpretation  can  be  given  to  the  undefined  symbols  which 
satisfies  P2  .  .  Pn  but  not  Pi.  This  particular  interpretation 
constitutes  an  independence  example  for  Pi. 

A  delicate  issue  is  raised  here.  How  can  we  be  sure  that 
P2  .  .  Pn  are  really  satisfied  by  the  independence  example? 
It  is  clearly  necessary  that  this  example  be  contained  within 
the  scope  of  a  science  determined  by  postulates  which  are  both 
consistent  and  categorical.  Here  again  we  are  driven  from  one 
domain  to  another,  and  if  we  wish  to  show  unqualifiedly  the 
validity  of  an  independence  example,  we  must  be  led,  as  before, 
back  to  the  number  system. 

We  must  make  special  mention  of  the  method  of  so-called 
vacuous  satisfaction,  which  is  often  used  in  setting  up  inde- 
pendence examples.  Suppose  we  have  an  axiom  to  the  effect 
that  "two  distinct  points  determine  a  straight  line"  and  we 
are  considering  the  geometry  of  a  single  point.  Here  there  is  no 
distinct  second  point,  but  we  say  that  this  unit  space  has  the 
property  that  every  two  points  determine  a  straight  line.  The 
axiom  is  vacuously  satisfied  by  a  non-existent  point  and  a  non- 
existent line.  Now  the  curious  part  of  the  reasoning  involved 
is  this, — if  the  contradictory  axiom  "two  points  never  determine 
a  straight  line"  were  under  consideration,  this  would  also  be 
vacuously  satisfied  by  the  space  of  a  single  point.  Now  this  is 
somewhat  of  a  departure  from  the  attitude  of  naive  common 
sense  and  we  must  be  prepared  to  admit  that  accepted  mathe- 
matical method  involves  to  some  extent  the  logical  doctrine 


58  The  Notion  of  Number  and  the  Notion  of  Class 

that  a  false  proposition  implies  any  arbitrary  proposition,  and 
its  analogues  for  classes. 

The  denoting  terms  used  in  mathematics  merit  especial  atten- 
tion. Let  us  take  a  familiar  example  from  elementary  analytic 
geometry.  Suppose  we  wish  to  deduce  the  equation  of  a  curve. 
We  must  first  know  how  that  curve  is  generated  by  conditions 
on  a  representative  point;  we  say  the  curve  is  the  locus  of  points 
such  that  this  condition  is  satisfied.  The  coordinates  of  this 
general  point  which  typifies  every  point  on  the  locus  are  taken 
to  be  (x,y)  and  it  is  required  to  connect  x  and  y  by  a  functional 
relation  determined  by  a  given  condition  on  the  general  point. 
First  we  have  the  condition,  second  the  general  element  and 
then  the  equation,  the  collection  of  particular  points  on  the 
curve.  We  derive  this  equation  by  a  significant  procedure. 
We  single  out  a  constant  point  P  which  is  any  definite  point 
on  the  locus  and  assume  its  coordinates  to  be  (xi,yi).  From 
the  condition  on  the  general  point,  we  arrive  at  a  functional 
relation  connecting  Xi  and  yi ;  and  then,  since  P  was  any  definite 
point  on  the  curve,  we  replace  xi  and  yi  by  x  and  y,  giving  the 
equation.  Now  it  might  be  said  that  the  point  (x,y)  is  really 
any  point  on  the  locus,  and  stands  for  each  particular  one;  so, 
since  (xi,yi)  also  was  any  point,  it  must  be  identical  with  (x,y). 
This  contention  has  only  superficial  and  verbal  plausibility. 
The  two  expressions,  considered  as  denoting  terms,  must  be 
carefully  differentiated.  In  the  first  place,  (x,y)  is  not  a  con- 
crete, particular  point  at  all.  If  the  expression  is  permitted, 
it  is  an  abstract  point  which  satisfies  the  common  condition, 
generates  the  points  of  the  curve  but  is  not  one  of  them. 
P(xi,yi),  on  the  other  hand,  is  one  of  the  points  of  the  curve, 
although  not  a  specifically  designated  one.  In  the  case  of  P, 
a  set  of  points  exists  which  are  on  the  curve  and  which  do  not 
include  P.  Where  the  typical  or  generating  point  is  considered, 
no  such  set  exists;  a  collection  of  points  independent  of  its  gener- 
ator would  be  self-destructive.  Whatever  can  be  proved  of 
the  general  term  is  certainly  true  of  P;  but  the  converse  does 
not  follow.  We  distinguish  therefore  between  the  indefinite 
any  (in  the  sense  of  the  variable),  any  definite  member,  the 
condition  given  at  the  start,  which  may  be  regarded  as  the  predi- 
cate common  to  the  points  on  the  locus  and  the  curve  as  a  single 
whole  regarded  as  the  range  of  the  variable  point.     We  have 


The  Notion  of  Class  59 

also  concluded  that  mathematical  reasoning  is  concerned  primar- 
ily with  words  considered  as  mere  symbols  or  marks  and  their 
possibilities  of  interpretation. 

§1.     The  Mathematical  Universe 

If  we  adhere  to  this  semi-verbal  character  of  accepted  mathe- 
matical reasoning,  it  is  possible  to  take  a  somewhat  broader 
view  of  the  method  and  content  of  mathematics  than  has  been 
the  fashion  among  the  logisticians.  Ordinary  experience  pro- 
vides us  with  a  vast  array  of  propositions  (taken  in  the  verbal 
sense).  With  how  these  propositions  are  acquired  we  are  not 
concerned ;  suffice  it  that  they  are  there, — the  objective,  common 
property  of  those  with  whom  logical  argument  is  possible.  Where 
there  is  no  such  common  store  of  propositions,  no  logic,  no  code 
of  agreement  or  disagreement  is  possible.  With  a  native  of  the 
South  Sea  Islands,  for  example,  no  logical  disputation  is  possible. 

This  array  of  propositions  as  presented  by  individual  experi- 
ence, common  sense,  and  inductive  science  is  undeniably  loose 
and  rambling  in  its  structure.  We  have  the  same  word  serving 
a  variety  of  uses  and  we  feel  that  between  such  propositions  as 
"2+2  =  5"  and  "the  present  King  of  France  is  bald"  there  is 
absolutely  no  connection.  Now  from  our  point  of  view,  the 
function  of  logical  method  is  just  this, — to  arrange  the  heterogen- 
eous concatenation  of  propositions  in  a  systematic  order;  to 
separate  the  disconnected  propositions  into  different  universes 
of  discourse. 

The  postulates  of  any  particular  code  of  logic  constitute  a 
specific  definition  of  a  systematic  universe  of  discourse;  and  the 
failure  of  any  part  of  an  agreed-upon  code  is  taken  to  be  an 
evidence  that  the  limits  of  the  universe  in  question  have  been 
overpassed.  For  example,  the  propositions  "A  is  blue,"  "A 
is  red,"  "A  exists,"  where  red  and  blue  are  taken  to  be  contra- 
dictory, cannot  be  contained  in  the  same  systematic  universe. 
We  conclude  that  the  first  and  second  must  be  taken  in  discon- 
nected universes,  psychologically  speaking,  from  different  points 
of  view  and  revise  them  thus,  to  put  them  in  the  same  universe 
"A  is  blue  at  time  x,"  "A  is  red  at  time  y."  The  goal  of  logical 
method,  in  general,  is  an  exact  and  accurate  statement  and 
analysis  of  our  rudimentary  and  vague  notion  of  a  systematic 
universe  of  discourse. 


60  The  Notion  of  Number  and  the  Notion  of  Class 

Some  such  theory  of  the  universe  of  discourse  is  implicitly 
presupposed  by  any  logic  and  may  be  summed  up:  "Any  prop- 
osition which  violates  the  logical  canons  can  be  split  up  into  a 
complex  of  propositions  to  be  taken  in  different  universes." 
The  code  must  at  all  costs  be  preserved.  Puns  must  be  handled 
in  this  way,  for  instance.  We  are  not  allowed  to  attribute 
existence  to  that  which  has  contradictory  predicates  (as  in  the 
case  of  blue  and  red)  but  are  forced  to  divorce  the  predicate 
if  we  wish  the  existence  to  remain.  Now  existence  is  not  a 
quality,  like  extension,  magnitude,  or  color.  The  size  or  color 
of  an  object  may  well  differ  in  different  universes,  but  it  has 
or  has  not  existence  in  a  given  universe.  In  the  sphere  of  a 
lecturer  on  Modern  Drama,  Hedda  Gabler  has  just  as  real  and 
objective  an  existence  as  anything  you  please,  while  to  a  chemist 
she  is  relegated  to  the  domain  of  the  round-squares  and  the 
chimaeras.  Any  word  which  violates  the  law  of  contradiction 
when  the  two  contradictory  propositions  are  united  in  a  common 
universe  cannot  have  existence  attributed  to  it. 

From  the  mere  absence  of  contradiction,  however,  existence 
is  not  to  be  inferred.  A  nonsense  word  can  be  given  existence 
in  a  universe,  but  does  not  possess  existence  of  its  own  accord 
unless  this  is  implied  by  accepted  postulates.  Hence  the  pres- 
ence of  the  existence  postulates  in  a  mathematical  system. 
They  give  a  beginning  to  the  collection  of  known  existents, 
which  is  added  to  by  use  of  the  other  postulates.  Roughly 
speaking,  existence  is  nothing  more  nor  less  than  membership 
in  a  certain  class.  Following  Russell  (in  1903),  we  will  hold 
that  words  which  do  not  exist  in  a  given  universe  have  being 
with  regard  to  that  universe.  Any  definite  word  has  being  with 
reference  to  any  definite  universe  we  choose  to  add  it  to,  and,  it 
may  be,  existence  in  some.  The  round-square  has  being  in 
the  universe  of  metrical  geometry;  it  may  have  existence  where 
"square"  is  interpreted  in  the  sense  of  "public  square"  as  the 
space  between  intersecting  streets.  Thus  a  public  square  which 
is  round  is  not  a  paradox  if  we  find  the  right  universe.  Obviously 
this  new  universe  and  geometry  are  disconnected,  at  least  so 
far  as  the  term  "square"  is  concerned. 

We  can  see  now  just  what  mathematicians  mean  when  they 
say  that  along  with  their  particular  axioms  they  assume  the 
postulates  of  logic.     They  mean  that  the  universe  of  discourse 


The  Notion  of  Class  61 

with  which  they  are  about  to  deal  is  a  systematic  universe,  subject 
to  the  law  of  contradiction. 

Assertion  is  to  propositions  what  existence  is  to  terms  not 
a  quality  or  attribute  but  containing  an  essential  reference  to 
the  universe  of  discourse.  An  asserted  proposition  occurs  as  a 
unit  factor  in  an  implication  and  can  always  be  dropped.  Axioms 
of  a  science  are  the  fundamental  asserted  propositions  of  the 
universe  they  define,  and  their  assertion  is  not  absolute,  but 
relative.  Assertion  is  not  the  same  as  truth,  which  in  fact  is  a 
highly  controversial  idea  and  seems  to  involve  notions  extraneous 
to  pure  logic.  When  we  consider  "A  is  B"  as  a  proposition, 
it  may  imply  its  contradictory  in  one  universe  and  be  implied 
by  it  in  another.  In  the  first  case  "A  is  not  B"  is  asserted,  in 
the  other  "A  is  B."  Propositions  which  are  implied  by  their 
contradictories  are  unit  propositions  with  respect  to  the  particu- 
lar universe;  those  which  imply  their  contradictories  are  null 
propositions. 

One  of  our  most  fundamental  demands  that  we  make  upon 
a  systematic  universe  is  that  with  regard  to  it  the  same  prop- 
osition cannot  be  both  unity  and  null.  When  such  a  state  of 
affairs  occurs  we  have  the  choice  of  two  alternatives:  (i)  either 
we  assume  that  somehow  the  unit  part  and  the  null  part  of  the 
contradiction  occur  in  different  universes  and  the  proposition 
cannot  be  asserted  without  some  change  which  expresses  this 
fact,  or  (2)  the  axioms  of  our  logic  are  insufficient  to  define  a 
truly  systematic  universe  and  must  be  revised  to  meet  the 
situation.  One  or  both  of  these  methods  must  be  adopted  in 
dealing  with  Russell's  paradoxes. 

§2.     The  Internal  Structure  of  a  Universe 

We  have  in  any  universe  words  which  play  the  part  of  con- 
ditions or  predicates.  Other  symbols  satisfy  these  conditions 
Conditions  which  are  satisfied  by  one  term  are  concepts;  by 
couples  with  sense,  relations;  by  trios  with  betweenness,  operations 
or  triadic  relations.  The  collection  of  symbols  which  satisfy 
a  condition  P,  considered  as  forming  a  single  whole,  constitute  the 
range  of  P. 

Suppose  that  within  some  systematic  universe  we  have  a  well- 
defined  concept  C.  Its  range  will  be  said  to  constitute  a  connected 
universe  of  discourse  if  any  definite  symbol  satisfying  C  occupies 


62  The  Notion  of  Number  and  the  Notion  of  Class 

only  one  place  with  regard  to  the  relation  of  satisfaction.  Thus 
if  A  satisfies  C,  A  satisfies  B,  C  satisfies  A;  the  A  of  the  first  two 
propositions  is  not  in  the  same  connected  universe  as  the  A  of 
the  last.  A  term  therefore  cannot  satisfy  itself,  unless  we  specify 
by  some  notation  that  the  first  term  of  the  relation  is  in  a  different 
universe  from  the  second.  The  same  restriction  holds  upon  the 
range  of  A;  it  cannot  satisfy  A  without  some  specification.  "Soc- 
rates, the  philosopher,  is  human"  and  "Human  is  a  word  having 
five  letters"  are  propositions  occuring  in  different  connected 
universes. 

Russell's  conception  of  the  absolutely  unrestricted  variable 
does  not  appear  to  be  needed.  He  says  that  in  " 'x  is  a  man' 
implies  'x  is  a  mortal,'"  it  is  only  a  vulgar  prejudice  in  favor 
of  true  propositions  which  restrains  us  from  substituting  bicycles 
and  teaspoons  for  x.  This  may  be  doubted;  and  from  our 
(intensional)  point  of  view  what  we  assert  in  the  "formal  impli- 
cation" in  question  is  a  relation  between  the  concepts  "man" 
and  "mortality"  which  we  may  call  the  relation  of  inclusion. 
The  so-called  "apparent"  variable  appears  to  be  a  confusion 
in  terms. 

Where  we  would  say  "for  all  values  of  x,  x  satisfies  C,"  we 
say  "C  is  a  unit  concept";  "for  no  values  of  x,  x  satisfies  C"  or 
"for  all  values  of  x,  x  satisfies  not-C," — "C  is  a  null  concept." 
If  C  is  a  unit  concept,  not-C  is  null,  and  vice  versa.  "  For  some 
values  of  x,  x  satisfies  C"  becomes  "C  is  not  null"  and  "for 
some  values  of  x,  x  satisfies  C," — "C  is  not  unity." 

To  express  relations  between  concepts  we  make  use  of  denot- 
ing terms,  which  together  with  concepts  form  denoting  com- 
plexes. Where  C  is  a  condition  and  x  is  the  variable  term  which 
stands  for  any  term  of  the  range  of  C,  for  this  variable  term 
we  use  the  denoting  complex  "every-C."  If  x  is  a  fixed  term 
but  not  specified  by  particular  properties,  we  replace  it  by 
"any-fixed  C." 

Where  "every-B  satisfies  A"  we  say  that  the  range  of  B  is 
included  in  the  range  of  A;  similarly  for  "any  fixed-B  satisfies 
A." 

In  the  term  "any-fixed"  no  particular  element  is  understood 
to  be  specified;  we  have  something  analogous  to  the  constant 
in  indefinite  integration  which  can  be  said  to  vary,  but  is  none 
the  less  a  constant  and  not  a  variable.     There  is  an  ambiguity 


The  Notion  of  Class  63 

about  a  parameter  which  is  distinct  from  the  essential  ambiguity 
of  the  variable  and  which  is  essential  to  mathematical  usage. 
Consider  ax+by+c=o.  For  fixed  values  of  a,b,c  we  have  a 
definite  equation;  taking  into  account  the  ambiguity  of  the 
parameters  we  see  that  we  have  a  triply  infinite  set  of  equations. 
In  this  discussion  x  and  y  are  regarded  to  maintain  a  placid 
and  invariant  state  of  variability  if  we  may  be  pardoned  the 
paradox;  a,b,c  are  seen  to  vary  in  different  ways,  though  always 
remaining  constants.  In  the  case  of  a  concept  C  with  only  a 
finite  number  of  terms  Si  .  .  .  Sn,  "every-C"  denotes  Si 
and  denotes  S2  .  .  .  and  denotes  Sn,  "any-fixed-C"  denotes 
Si  or  S2  or  .  .  .  Sn.  The  case  of  "a  selected-C"  is  quite 
different.     Here  we  have  a  true  constant  independently  defined. 

The  range  of  C  is  to  be  taken  very  much  as  "all-C's"  consid- 
ered collectively.  Following  Frege  we  arbitrarily  assert  that 
the  denoting  complex  "the-C"  is  equivalent  to  the  range  of  C. 
Thus  "the  positive  integer  between  one  and  three"  is  "two," 
considered,  not  as  individual,  but  as  sole  member  of  the  range. 
"The  square  root  of  four"  is,  without  previous  restriction,  the 
collected  unity  of  such  roots;  "the-lion"  expresses  the  range  of 
the  concept  lion.  "The  round-square"  (in  a  geometrical  uni- 
verse) denotes  a  null-range.  Thus  every  description  is  given 
an  equivalence. 

Any  symbol  which  satisfies  a  null  condition  can  be  imported 
into  the  range  of  any  condition  we  please;  a  null  range  or  any 
part  of  it  is  included  in  any  fixed  range,  null  or  not.  Thus  if  we 
have  one  point  P  and  one  null  point  P1,  P  and  P1  can  be  regarded 
as  determining  a  straight  line,  or  not,  at  pleasure.  In  actual 
practice  the  activity  of  the  logician  contributes  something. 
The  existent  members  of  a  range  are  considered  to  be  there, 
given;  these  non-existents  are  only  potentially  present  and  are, 
in  effect,  put  there  by  the  voluntary  choice  of  the  logician  who 
chooses  to  make  them  satisfy  this  rather  than  the  contradictory 
condition.  Logic  actually  contributes  something;  it  is  not  a 
mere  passive  spectator  in  its  own  development. 

The  calculus  of  ranges  is  precisely  that  of  Peano's  calculus 
of  class  concepts  (which  he  looks  at  from  the  standpoint  of 
extension),  if  we  allow  the  term  inclusion  to  be  used  in  such  a 
way  that  the  concept  C  is  included  in  Cl,  if  every  C  satis- 
fies O. 


64  The  Notion  of  Number  and  the  Notion  of  Class 

§3.     Definition,  Selection,  and  Critical  Questions 

We  will  say  that  a  condition  C  is  defined  with  reference  to  a 
systematic  universe,  if  every  symbol  which  satisfies  C  and  not-C 
is  a  non-existent.  If  the  universe  is  also  connected,  C  is  said  to 
be  well-defined.  Any  well-defined  condition  not-null  is  an  exis- 
tent condition.  If  C  is  an  existent,  the  range  of  C,  every-C,  and 
any-fixed-C  exist. 

Any  existent  in  a  systematic  universe  is  said  to  possess  an 
indication  with  respect  to  that  universe;  if  the  universe  is  con- 
nected,   it    possesses   a   unique   indication.     Any    non-existent 
possesses  a  unique  indication  (since  it  indicates  nothing  at  all). 
Limiting  assertions  to  a  connected  universe  is  no  more  nor  less 
than  an  objective  rendering  of  the  traditional  dogma  that  in 
every  argument  the  terms  used  must  have  a  constant  "mean- 
ing."   The  intrusion  of  subjective  elements  in  "meaning"  seems 
unavoidable,   but  by   making   mathematical   reasoning   purely 
verbal,  the  psychological  factors  are  reduced  to  a  minimum. 
Looked  at  from  the  mathematical  point  of  view  what  we  have 
said  amounts  to  this, — given  a  set  of  postulates,  general  and 
existential,  then  for  every  demonstrated  existent  in  the  universe 
determined  by  the  postulates,  there  is  one  and  only  one  inter- 
pretation for  a  given  interpretation  of  the  undefined  symbols. 
For  any  demonstrated  non-existent  we  can  substitute  nothing. 
"Meaning,"  therefore,  becomes  a  term  relative  to  the  universe. 
The  more  systematic  the  universe,  the  less  intrusion  of  the 
subjective.     In  a  hap-hazard  universe  where  only  a  few  funda- 
mental laws  hold,  psychology  may  indeed  run  riot.     The  prog- 
ress of  objective  science  and   mathematics  is  essentially   the 
analysis  of  such   loosely  put   together  universes   into  several 
systematic  ones.     Consider  the  proposition,   "Walt  Whitman 
is  a  poet."     In  a  conversational  universe,  individual  opinion  of 
necessity  intrudes  and  we  assent  or  dissent  according  to  our 
individual  notions  of  what  "true  poetry"  may  be  or  not  be. 
"Poet"  is  a  term  which  does  not  possess  a  unique  indication 
since  the  universe  is  unsystematic.     However,  if  we  go  about 
the  matter  logically  and  scientifically,  we  define  poetry  in  objec- 
tive terms  and  give  the  term  a  unique  indication. 

We  must  now  take  up  our  postponed  discussion  of  the  denot- 
ing complex  "a-selected"  which  is  roughly  equivalent  to  "an 
independently  defined."     What  we  mean  by  this  is — given  a 


The  Notion  of  Class  65 

concept  (not  null)  is  there  any  means  of  defining  a  symbol  which 
satisfies  it?  Any  concept  whose  range  is  included  in  the  range 
of  the  concept  "positive  integer"  does  provide  such  a  means. 
Here  the  selective  function  is  "the  first,"  since  every  collection 
of  positive  integers  must  have  a  first  term.  Consequently  if 
0  =  "positive  integer"  and  C2  satisfies  Ci,  a-selected-Ca  exists. 
(C  =  "sub-condition  of  C.) 

Any  condition  C  where  a-selected-C  exists  is  said  to  be  selec- 
tive. Any  condition  C  whose  associated  condition  C  (of  included 
conditions)  is  uniformly  selective  is  said  to  be  a  Zermelo  con- 
dition. 

We  must  now  consider  two  logical  axioms  which  have  to  do 
with  the  notion  of  selection. 

(1)  Every  condition  which  is  well-defined  and  not-null  is 
selective.     This  will  be  called  the  selective  axiom. 

(2)  Every  well-defined  condition  not  null  is  a  Zermelo  con- 
dition.    This  is  the  so-called  Zermelo  postulate. 

First  we  may  ask,  can  it  be  proved  directly  that  being  selec- 
tive follows  from  being  well-defined?  Let  us  suppose  a  concept 
C  contained  in  an  infinite  universe,  where  we  can  name  as  many 
terms  as  we  please.  Is  there  a  method  by  which  we  can  name 
a  definite  symbol  satisfying  C?  If  we  take  term  after  term 
of  the  universe,  it  is  determinate  in  each  and  every  case  whether 
it  satisfies  C  or  not-C,  but  we  may  have  the  ill  luck  to  hit  upon 
those  which  satisfy  not-C.  Since  we  are  unable  to  specify 
all  the  terms  of  the  universe,  no  proof  seems  possible  on  the 
basis  of  the  bare  supposition  that  C  is  well-defined. 

The  truth  of  the  axiom  may,  in  point  of  fact,  be  sincerely 
doubted  and  Russell  actually  cites  "all  products  of  two  integers 
which  never  have  been  and  never  will  be  thought  of  by  any 
human  being"  as  a  well-defined  condition  not  null  of  which  no 
instances  can  be  given, — hence  non-selective.  Physical  objects 
and  other  people's  minds  are  other  such  instances,  he  contends. 

Let  us  consider  abstractly  what  must  be  done  to  demonstrate 
that  a  given  well-defined  condition  P  should  be  non-selective, 
i.  e.,  a-selected-P  does  not  exist.  Since  from  the  fact  that  P 
is  well-defined  it  follows  that  any  fixed  symbol  we  mention 
determinately  satisfies  P  or  not-P,  then  any  such  symbol  must 
satisfy  not-P  if  we  are  to  show  P  to  be  non-selective.  Any 
demonstrated  existent  in  the  universe  must  be  shown  to  satisfy 


66  The  Notion  of  Number  and  the  Notion  of  Class 

not-P.  Therefore,  if  P  is  a  known  existent,  we  must  show  that 
P  satisfies  not-P  and  this  cannot  be  done  without  departing 
from  our  connected  universe,  which  cannot  be  allowed  since  P 
was  supposed  to  be  well-defined.  Such  examples  as  Russell's  con- 
tain a  latent  vicious  circle  fallacy.  Take  for  example,  the  familiar 
"proof"  of  idealism  that  all  entities  are  ideas.  Everything  that 
can  be  mentioned  is  an  idea,  and  you  cannot  show  an  entity  to 
exist  without  mentioning  it,  so  all  entities  are  ideas.  The  prop- 
osition to  be  proved,  however,  is  certainly  an  entity  and  to 
show  that  it  is  mentioned,  it  is  necessary  to  change  from  one 
point  of  view  to  another  and  so  to  use  "mentioned"  in  a  different 
sense.  Such  notions  as  mentioning,  thinking  of,  prove  nothing 
about  all  entities. — The  other  examples  such  as  other  people's 
minds  and  physical  objects  are  more  philosophical  scandals 
than  anything  else,  and  in  the  sense  in  which  Russell  uses  the 
terms,  their  existence  may  be  denied  without  any  great  display 
of  radicalism. 

Can  we  conclude  from  this  that  we  are  justified  in  assuming 
the  selective  axiom?  Here  is  a  cross  road  where  two  schools 
part  company.  It  is  possible  to  steer  a  safe  and  cautious  course 
not  denying  the  axiom  but  not  assuming  it  and  when  necessary 
actually  determining  the  needed  selective  function.  The  great 
majority  of  mathematicians,  however,  implicitly  assume  that 
every  well-defined  condition  is  selective.  A  point  which  appears 
to  justify  this  assumption  is  the  nature  of  indirect  proof  (reductio 
ad  absurdum).  We  start  with  a  condition  P  and  we  seek  to 
show  not-P  to  be  null.  We  take  (let  us  suppose)  a-definite 
not-P  and  show  that  it  has  contradictory  properties.  From  this 
we  conclude  that  not-P  is  null  since  a-selected  not-P  does  not 
exist.  If  the  selective  axiom  is  false  nothing  can  be  concluded 
and  all  indirect  proof  fails. 

Although  the  necessary  distinctions  are  very  difficult  and 
cannot  be  made  with  any  great  degree  of  confidence,  it  appears 
that  a  case  may  be  made  in  rebuttal.  In  all  but  a  very  small 
percentage  of  proofs  by  reductio  ad  absurdum,  the  member  of 
not-P  being  definite  has  nothing  to  do  with  the  argument. 
Consequently  for  ordinary  mathematical  purposes  instead  of 
a-selected  not-P  we  take  any-fixed-not-P  and  show  that  it  does 
not  exist.  This  is,  of  course,  a  valid  proof  since  we  assumed 
that  if  P  is  well-defined  and  not  null,  any-fixed-P  exists.     For 


The  Notion  of  Class  67 

the  cases  where  definiteness  is  needed  we  either  reject  the  con- 
clusion, seek  a  direct  proof  or  take  the  conclusion  as  a  postulate 

The  Zermelo  postulate  has  gained  much  greater  notice  in 
mathematical  circles.  Cantor,  Bernstein,  Whitehead,  Schonfiies, 
and  many  other  prominent  mathematicians  have  accepted  it 
either  explicitly  or  implicitly  and  it  is  thought  that  a  refusal 
to  adopt  it  is  paradoxical  and  involves  a  sweeping  away  of 
"much  interesting  mathematics."  The  more  philosophic  writers 
on  the  foundations  of  Analysis,  such  as  Russell,  Poincare,  Hob- 
son,  and  Borel,  have  expressed  doubts  as  to  its  truth.  It  is 
contended  that  it  is  the  equivalent  of  an  infinite  number  of 
acts  of  choice  and  involves  all  the  difficulties  of  the  selective 
axiom  (which  is  a  particular  case)  with  additional  ones  of  its 
own.  The  postulate,  itself,  is  equivalent  to  the  selective  axiom 
and  either  of  (a)  every  condition  can  be  well-ordered  or  (b)  of 
any  two  cardinal  numbers  one  is  greater  than,  less  than,  or 
equal  to,  the  other.  It  is  preferable  to  prove  (a)  and  (b),  how- 
ever, if  the  postulate  is  justified,  since  they  are  essentially  mathe- 
matical in  character.  Before  discussing  the  postulate  critically, 
it  is  well  to  say  a  few  words  about  its  suppossed  self-evidence. 
This  is  brought  about  by  assuming  that  "any-fixed"  can  be 
regarded  as  a  selective  function  which  applied  to  any  definite 
collection  of  conditions  yields  an  equivalent  collection,  definite, 
of  members  of  the  conditions.  While  superficially  plausible 
there  is  no  reason  to  suppose  that  "any-fixed"  yields  anything 
definite  except  in  the  case  of  singular  conditions  (with  only  one 
term).     This  "self-evidence,"  therefore,  is  illusory. 

A  certain  uniformity  of  selection  is  what  is  guaranteed  by 
the  postulate.  The  conditions  from  which  we  are  to  choose, 
unless  finite  in  number,  must  possess  a  similarity  of  structure 
which  permits  the  condition  gained  by  the  selection  to  be  well- 
defined.  Consider  the  interiors  of  circles.  The  interiors  of 
mutually  exclusive  circles  obey  the  Zermelo  postulate,  since 
they  are  not  homogeneous  but  contain  an  element  bearing  a 
unique  relation  to  the  concept — the  center.  The  selective 
axiom  assumes  this  non-homogeneity  with  reference  to  each 
individual  condition. 

We  will  now  discuss  some  critical  points  with  especial  refer- 
ence to  the  paradoxes  and  the  interpretation  of  ranges.  The 
paradoxes  are,   we  hope,   successfully  avoided   by   the  notion 


68  The  Notion  of  Number  and  the  Notion  of  Class 

of  unique  indication;  to  have  a  unique  indication,  we  recall,  a 
symbol  must  be  contained  in  a  connected  universe.  Now 
Russell  has  shown  that  the  paradoxes  result  as  particular  cases 
of  the  following:  "Given  a  property  §  and  a  function  f  such  that 
if  (J>  belongs  to  all  the  members  of  u,  f(u)  always  exists,  has  the 
property  <j>  and  is  not  a  member  of  u ;  then  the  supposition  that 
there  is  a  class  w  of  all  terms  having  the  property  <J>  and  that 
f(w)  exists  leads  to  the  conclusion  that  f(w)  both  has  and  has 
not  the  property  <J>."  Here  f(u)  occurs  as  function  (condition) 
and  as  symbol  satisfying  (J>,  which  cannot  occur  in  one  connected 
universe,  without  some  notation  to  express  the  difference.  How- 
ever, if  the  notation  is  introduced,  the  paradox  vanishes. 

For  a  particular  example,  take  the  predicate  "not  predicable 
of  itself."  In  our  language  this  would  be  the  condition  "con- 
dition which  does  not  satisfy  itself."  Let  this  be  W.  Then 
if  A  satisfies  W,  A  satisfies  not-A  and  conversely.  So  if  W 
satisfies  W,  W  satisfies  not-W  and  conversely,  which  is  paradox. 
To  keep  the  argument  in  one  connected  universe,  however,  we 
must  give  our  terms  when  conditions  the  subscript  c  and  as 
symbols,  s.  All  we  get  now  by  our  substitution,  in  As  sat.  Wc  = 
A9  sat.  not-Ac  is,  if  Ws  satisfies  Wc,  Ws  satisfies  not-Ac  which 
has  nothing  to  do  with  the  case,  or  if  A  satisfies  Wc,  A  satisfies 
not-Wc  which  shows  that  Wc  is  not  well-defined.  The  supposed 
paradox  results  from  ambiguity. 

Burali-Forti's  interesting  ordinal  contradiction  merely  shows 
that  the  ordinal  number  of  all  ordinal  numbers  is  not  an  ordinal 
number  in  the  same  sense  (or  connected  universe)  as  its  mem- 
bers— thus  if  B  is  the  suspected  ordinal,  B>B  is  consistent 
because  in  the  one  case  it  is  condition  and  in  the  other  symbol. 
The  inequality  is  not  asserted  in  a  connected  universe  because 
there  the  set  of  ordinal  numbers  does  not  have  an  ordinal  number 
which  is  comparable  with  its  members.  The  class  of  all  classes 
and  cardinal  number  of  all  cardinal  numbers  involve  similar 
ambiguities,  where  ambiguity  is  understood  to  mean  not  any- 
thing subjective  but  lack  of  a  unique  indication,  as  we  have 
defined  the  term. 

Classes  are,  as  might  be  supposed,  dependent  upon  ranges 
and  our  original  query  "what  is  a  class?"  must  wait  for  the 
answer  to  "what  is  a  range?"  We  must  now  meet  Russell's 
objection  that  no  such  entity  as  a  Frege  range  is  comprehensible 


The  Notion  of  Class  69 

and  to  do  this  we  must  interpret  the  term  "range"  in  some  way 
consistent  with  experience.  We  recall  that  if  every-C  satisfies 
D  and  every-D  satisfies  C,  the  range  of  C  is  equivalent  to  the 
range  of  D;  if  A  satisfies  C,  it  belongs  to  the  range  of  C. 

The  clue  to  the  interpretation  of  ranges  must  be  found  in 
Frege's  all  too  brief  remarks  in  the  opening  paragraph  of  the 
"Grundgesetze."  Here  in  introducing  the  notion  of  a  function, 
he  comments  by  way  of  explanation  that  the  essence  of  a  func- 
tion is  not  in  the  mere  collection  of  values  which  satisfy  it,  but 
rather  in  their  "Zusammengehorigkeit."  Correspondingly  we 
may  say  that  a  range  possesses  a  coherency,  a  continuity  and 
a  homogeneity  which  a  loosely  put  together  heterogeneous  collec- 
tion does  not.  Each  member  of  a  range  is  an  example  of  the 
typical,  representative  general  term;  they  are  all  held  together 
by  the  fact  that  they  satisfy  the  same  condition. 

The  difference  between  a  range  and  a  collection  may  be  illus- 
trated by  the  classical  example  of  the  man  who  could  not  see 
the  forest  for  the  trees.  The  trees  represent  (from  his  point  of 
view)  a  collection  and  unless  known  to  be  finite  do  not  constiutte 
a  single  whole;  the  forest  represents  a  synthesis  and  each  par- 
ticular tree  is  indicated  by  "the  tree  of  the  forest."  The  analytic 
person  who  was  a  victim  of  this  strange  blindness  failed  to  see 
in  each  separate  tree  the  residuum  of  identity  which  it  has  in 
common  with  the  other  different  trees.  With  the  perception 
of  this  community  of  quality  comes  the  recognition  that  the 
collection  is  a  range — a  forest.  When  we  have  a  range  and 
perform  an  operation  on  a  general  term,  that  self  same  operation 
is  performed  on  the  different  terms  of  the  range.  If  x  is  the 
variable  tree  of  the  forest,  then  for  x  to  be  leafless  is  to  imply  a 
forest  denuded  of  its  foliage.  The  unity  of  a  range  is  not  the 
unity  of  an  additive  whole  but  more  that  of  an  organism  and  is 
based  upon  the  common  possession  of  an  identical  property. 
A  finite  collection  can  be  turned  into  a  range  by  the  addition 
of  singular  ranges.  An  infinite  collection  cannot.  Hence  we 
must  insist  that  mathematics  deal  only  with  ranges,  and  tolerate 
collections  only  as  potential  ranges. 

§4.     Classes,  Numbers,  and  the  Principle  of  Abstraction 

Let  us  assume  that  the  use  of  ranges  has  been  justified;  we 
have  still  not  attained  the  goal  of  this  whole  discussion, — the 
notion  of  class.     It  must  be  concluded  that  whereas  a  range  is 


yo  The  Notion  of  Number  and  the  Notion  of  Class 

something  concrete,  dependent  upon  its  particular  condition, 
a  class  is  independent  of  any  particular  condition  and  has  just 
as  much  to  do  with  any  fixed  one  of  a  set  of  formally  equivalent 
conditions  as  any  other;  it  is,  therefore,  not  concrete  in  the  sense 
that  a  range  is  concrete.  Whereas  we  found  a  range  to  depend 
upon  the  internal  homogeneity  of  the  symbols  satisfying  a  con- 
dition, a  class  depends  upon  the  external  homogeneity  of  ranges 
themselves,  in  short,  upon  the  principle  of  abstraction. 

Russell  is  enabled  to  dispense  with  the  principle  of  abstraction 
as  a  postulate  by  using  as  his  underlying  identity,  belonging 
to  the  class  of  terms  which  bear  to  one  another  the  isoid  (reflex- 
ive, symmetrical  and  transitive)  relation.  When  he  comes  to 
the  notion  of  class  itself  he  finds  himself  at  a  loss,  for  here  where 
the  use  of  the  principle  of  abstraction  would  be  most  advantage- 
ous, its  employment  is  impossible.  To  hold  that  the  definition 
of  class  is  that  a  class  is  the  class  of  formally  equivalent  conditions 
which  determine  it  would  be  a  vicious  circle  obvious  at  a  glance. 
The  principle  as  used  by  Russell  fails  utterly  to  take  care  of 
the  very  fundamental  notion  of  class. 

Against  his  use  of  it  in  general,  we  must  contend  that  although 
he  avoids  the  introduction  of  a  new  axiom,  this  advantage 
is  superficial  and  illusory,  (i)  If  the  existence  of  classes  is 
doubtful,  the  existence  of  the  entities  gained  by  the  abstraction 
is  also  doubtful,  since  they  are  classes.  Numbers,  since  defined 
by  this  process  become  meaningless,  and  arithmetic  falls  peril- 
ously near  to  nominalism.  (2)  The  principle  is  not  consistently 
employed,  as  has  been  very  clearly  pointed  out  by  Jourdain. 
The  definitions  of  cardinal  number  and  ordinal  number  go  well 
enough,  but  when  we  come  to  real  number — instead  of  defining 
a  real  number  as  a  class  of  coherent  classes  of  rationals,  as  the 
principle  clearly  indicates,  Russell  arbitrarily  defines  the  real 
number  as  one  particular  class,  a  segment.  This  inconsistency 
moves  Jourdain  to  inquire  whether  cardinal  numbers  could  not 
have  been  defined  in  a  similar  way,  and  he  in  fact,  proposes 
another  method  based  upon  this  new  principle  of  abstraction. 
Me  believes  that  it  is  possible  to  define  Zero  as  the  null-class, 
One  as  the  class  of  classes  whose  only  member  is  the  null  class, 
and  so-on, — substituting  for  the  Frege-Russell  "class  of  classes 
equivalent  to  the  given  class"  one  particular  member  of  this 
set.     Now  this  definition  in  no  way  helps  us  with  the  notion  of 


The  Notion  of  Class  71 

class  and  has  little  advantage  of  any  kind  to  recommend  it. 
It  does,  however,  tend  to  cast  considerable  doubt  upon  Russell's 
use  of  the  principle  of  abstraction. 

To  maintain  the  validity  of  the  principle  along  with  the 
unreality  of  classes  seems  mere  verbiage,  and  to  dispense  with 
its  use  altogether,  an  unnecessary  hardship.  We  propose,  there- 
fore, the  following  logical  postulate  which,  if  we  are  correct,  is 
sufficient  foundation  for  the  existence  of  classes  and  numbers 
and  in  fact  is  somewhat  analogous  to  Russell's  axiom  of  reduci- 
bility:  Given  a  non-singular  collection  of  entities,  between  any 
fixed  pair  of  which,  there  subsists  an  isoid  relation  R,  then  there 
exists  a  unique  well-defined  symbol  ARto  which  every  term  of  the 
collection  bears  the  relation  R,  but  which  is  distinct  from  any 
definite  term  of  the  collection.  This  abstract  entity  is  not  one 
of  any  collection  of  entities  having  the  relation;  it  is  exceptional 
and  not  on  a  par  with  them,  since  they  are  concrete. 

A  comparison  with  Cantor's  conception  of  abstraction  will 
make  this  postulate  clearer  and  more  plausible.  Cantor,  we 
recall,  abstracted  order  and  quality  from  a  group  and  gained 
the  cardinal  number.  Obviously  this  new  abstract  entity  he 
got  in  this  way  is  not  identical  with  any  of  the  concrete  groups, 
although  it  bears  to  them  a  very  curious  relation,  very  much 
as  the  general  term  of  a  condition  bears  to  any  particular  one. 
This  abstract  group  does  not  have  the  number;  it  is  the  number. 
It  bears  to  any  of  its  concrete  groups  the  relation  of  one-one 
correspondence,  and  is  a  new  entity,  gained  by  a  neiv  principle. 

Let  us  apply  our  postulate  to  the  problem  of  the  definition  of 
classes  and  numbers.  Ranges  may  have  the  relation  of  formal 
equivalence  (mutual  inclusion),  which  is  isoid.  By  our  postulate, 
therefore,  given  say  the  range  of  Ci  formally  equivalent  to  the 
range  of  C2,  etc.,  then  there  exists  an  abstract  range  or  class 
formally  equivalent  to  O  and  to  C2,  etc.,  but  not  identical  with 
any  definite  one  of  these  ranges.  Hence  if  A  satisfies  Ci,  it 
belongs  to  this  Class,  which  we  will  speak  of  as  the  class  of  sym- 
bols satisfying  any  fixed  one  of  the  conditions  whose  ranges  are 
formally  equivalent.  The  necessary  and  sufficient  condition  for 
the  existence  of  a  class  is  the  existence  of  two  well-defined  con- 
ditions whose  ranges  are  formally  equivalent.  One  is  not  enough 
because  the  symmetry  and  transitivity  of  the  relation  do  not 
appear;  in  fact  from  one  range,  there  does  not  seem  to  be  any 


72  The  Notion  of  Number  and  the  Notion  of  Class 

motive  for  abstracting.  We  require,  therefore,  that  before  the 
abstraction  occur,  at  least  two  entities  must  have  the  required 
relation  to  each  other. 

Classes  as  we  have  defined  them  possess  the  limitations  of 
ranges.  They  are  in  the  same  connected  universe  as  the  ranges 
they  are  equivalent  to  and  cannot  satisfy  the  conditions  of 
these  ranges  and  at  the  same  time  possess  members  without 
this  difference  being  symbolically  expressed.  The  paradoxes 
of  classes  are  analogous  to  those  we  have  considered  for  con- 
ditions. 

We  come  now  to  the  definition  of  abstract  cardinal  number. 
Let  us  suppose  numerical  equivalence  defined  as  Frege  has 
defined  it,  an  isoid  relation.  Then  given  a  collection  of  classes 
(not  ranges)  which  are  numerically  equivalent,  our  postulate 
gives  us  a  new  entity  more  abstract  than  the  classes  in  question 
but  equivalent  to  them — this  is  the  cardinal  number. 

Given  a  definite  class  A  and  the  relation  of  numerical  equiva- 
lence, then  providing  A  has  been  abstracted  from  a  selective 
condition,  then  the  cardinal  number  of  A  exists.  For  the  selec- 
tive member  of  A  can  be  replaced  by  something  else  at  pleasure 
and  by  this  method  we  get  a  new  class  A1,  which  is  in  (1,1) 
correspondence  with  A,  satisfying  the  hypothesis  of  the  prin- 
ciple of  abstraction.  If  Po  and  Po1  are  null  conditions,  their 
ranges  are  formally  equivalent  and  determine  the  null-class 
which  has  no  existent  members. 

A  singular  condition  can  be  defined  in  logical  terms  and  One 
is  the  cardinal  number  gained  by  abstracting  from  singular 
conditions.  Now  if  One  exists,  a  singular  condition  C  to  which 
it  applies  also  exists  and  by  considering  these  as  symbols  we 
get  an  additive  range,  selective,  which  yields  by  abstraction 
the  cardinal  Two.  A  cardinal  R  is  said  to  be  the  immediate 
successor  of  another  cardinal  S  provided  R  applies  to  a  condition 
P  which  contains  a  singular  condition  Pi  such  that  the  cardinal 
number  of  P-Pi  is  S.  Two  is  therefore  the  immediate  successor 
of  One.  Three  exists,  by  considering  Two,  One,  and  C  as  sym- 
bols and,  in  fact,  we  can  proceed  as  far  as  we  like  by  this  method, 
changing  the  connected  universe  at  each  step. 

It  does  not  appear,  however,  that  in  any  given  connected 
universe  every  number  has  a  successor,  for  the  universe  has 
been  changed  at  each  step.     Can  this  gap  be  filled  by  Frege's 


The  Notion  of  Class  73 

method  of  introducing  Zero?  We  must  remember  that  the 
null  class  contains  no  existent  members  and  that  it  is  unique, — 
there  may  be  many  null  ranges  and  null  conditions  but  there  is 
only  one  null  class.  The  necessary  and  sufficient  condition 
that  this  null  class  should  give  rise  to  a  cardinal  number  Zero 
in  the  way  that  a  singular  class  yields  One  is  that  it  be  selective. 
Now  can  we  possibly  hold  that  the  null-class  is  selective?  It  is 
hard  to  see  how  such  a  contention  can  be  maintained  with  any 
show  of  plausibility, — yet,  if  we  deny  it,  Zero  is  not  a  cardinal 
number  in  the  sense  that  One,  Two,  Three,  etc.,  are  cardinal 
numbers.  There  are,  however,  certain  psychological  reasons 
for  believing  that  this  is  the  case. 

We  seem  to  be  confronted  with  a  choice  of  alternatives:  (a) 
we  assume  the  selective  character  of  the  null  class  and  by  the 
method  of  logical  activity  (including  the  null  class  in  particular 
classes)  show  that  every  number  has  a  successor,  or  (b)  we  start 
the  number  series  with  One  and  substitute  for  the  use  of  Zero, 
an  axiom  to  the  effect  that  every  number  has  a  successor. 

It  is  possible  to  hold  that  in  either  case  an  assumption  has 
been  made  which  differentiates  Cardinal  Arithmetic  from  Pure 
Logic.  The  notion  of  logical  activity  enables  us  to  hold  that 
Logic  is  the  class  of  all  propositions  gained  without  placing 
non-existents  in  arbitrary  classes.  However  (a)  seems  to  depend 
upon  the  null-class  in  an  objectionable  way  and  there  seems 
to  be  much  better  reason  for  adopting  the  second  of  the  two 
possibilities  and  holding  that  the  axiom  (every  number  has  a 
successor)  marks  the  dividing  line  between  Logic  and  Arith- 
metic. 

In  this  we  are  to  some  extent  confirmed  by  the  verdict  of 
others.  This  was  one  of  Kerry's  objections  to  Frege,  and  it 
was,  in  germ,  the  keynote  of  Poincare's  vigorous  assaults  upon 
logistic.  He  believed  that  a  new  type  of  reasoning  was  involved 
in  theoretical  arithmetic,  that  mathematical  induction  repre- 
sented a  distinct  type  of  intuition,  a  separate  faculty  of  the  mind. 
These  results  he  gained  more  by  introspection  than  by  abstract 
reasoning,  and  we  cannot  accept  subjective  reasons  for  such 
theories.  Couterat  and  Russell  were  undoubtedly  in  the  right 
when  they  emphasized  the  objective  character  of  logistic.  Trans- 
lated into  objective  terms,  however,  Poincare's  theory  is  quite 
different  and  is  equivalent  to:  A  new  indefinable  or  a  new  axiom 


74  The  Notion  of  Number  and  the  Notion  of  Class 

must  be  introduced  into  logic  before  its  indefinables  and  axioms 
are  sufficient  to  yield  cardinal  arithmetic.  This  is  substantially 
our  own  conclusion. 

Russell  has  himself  implicitly  given  up  the  battle  although 
the  admission  of  defeat  is  not  so  clear  as  might  be  wished.  He 
says  in  the  "Principia" — "Some  of  the  properties  which  we 
expect  inductive  cardinals  to  possess  .  .  can  only  be  proved 
by  assuming  that  no  inductive  cardinal  is  null.  .  .  This 
amounts  to  the  assumption  that  in  any  fixed  type,  a  class  can 
be  found  having  any  assigned  inductive  number  of  terms.  .  . 
This  assumption  .  .  will  be  adduced  as  a  hypothesis  when- 
ever it  is  relevant.  It  seems  plain  that  there  is  nothing  in  logic 
to  necessitate  its  truth  or  falsehood  and  that  it  can  only  be 
legitimately  believed  or  disbelieved  on  empirical  grounds. 

Since  this,  in  effect,  concedes  the  point  at  issue,  we  need  not 
discuss  the  matter  further.  Poincar6  must  be  adjudged  the 
victor,  and  the  sweeping  assertion  of  1903 — "The  fact  that  all 
mathematics  is  symbolic  logic  is  one  of  the  greatest  discoveries 
of  our  age;  and  when  this  fact  has  been  established,  the  remainder 
of  the  principles  of  mathematics  consists  in  the  analysis  of  sym- 
bolic logic  itself"  appears  to  stand  in  need  of  considerable  revi- 
sion. The  dividing  line  between  Logic  and  Mathematics  is 
the  axiom  that  every  number  has  a  successor;  roughly  speaking 
a  new  notion  is  introduced — "aw d  so  on." 


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